Physics-Constrained Data-Driven Variational Method for Discrepancy Modeling

Abstract

This paper presents a data-driven discrepancy modeling method that variationally embeds measured data in the modeling and analysis framework. The proposed method exploits the residual between the first-principles theory and sensor-based measurements from the dynamical system, and it augments the physics-based model with a variationally derived loss function that is comprised of this residual. The method was first developed in the context of linear elasticity (Masud and Goraya, J. Appl. Mech. 89 (11), 111001 (2022)) wherein the relation between the discrepancy model and loss terms was derived to show that the data embedding terms behave like residual-based least-squares regression functions. An interpretation of the stabilization tensor as a kernel function was formally established and its role in assimilating a-priori knowledge of the problem in the modeling method was highlighted. The present paper employs linear elastodynamics as a model problem where the Data-Driven Variational (DDV) method incorporates high-fidelity data into the forward simulations, thereby driving the problem with not only the boundary and initial conditions, but also by measurement data that is taken at only a small subset of the total domain. The effect of the loss function on the time-dependent response of the system is investigated under a variety of loading conditions and model discrepancies. The energy and Morlet wavelet analyses reveal that the problem with embedded data recovers the energy and the fundamental frequency band of the target system. Time histories of strain energy and kinetic energy of a cantilever beam undergoing damped oscillations are recovered by including known data in an undamped model to highlight the data-driven discrepancy modeling feature of the method under the combined effect of parameter and model discrepancy.

1. Introduction

Physics-based modeling methods employ partial differential equations (PDEs) that are derived from first principles, conservation laws, or simple empirical observations, and they serve as mathematical idealizations of the physical systems. There are many complex phenomena for which quantitative analytic descriptions may not be exactly possible and the governing equations for them are only incomplete representations of the underlying physics. In these cases, PDE-based models serve as surrogate models with potential missing features in the overall physics that can be attributed to (i) incomplete knowledge about material behavior (parameter discrepancy), (ii) incorrect information about boundary conditions (model inadequacy), (iii) inexact information about the geometry of the domain, (iv) a-priori homogenization or reduced-order modeling of the system where physics may be oversimplified, and (v) incorrect structure of the mathematical model where important terms may be missing or erroneous terms may be present (model discrepancy). These deficiencies in the physics-based models result in discrepancies in the modeled response, as compared to observations and sensor-based measurements of the system, thereby revealing the approximate nature of the equations.

In recent years, data-driven methods and machine learning techniques are extensively being used to augment the modeling capability of physics-based models [1]. These include kernel-based regression and support vector machines [2, 3], probabilistic machine learning methods [4, 5], data-driven methods using sparse observations [6, 7], and physics-informed neural networks [8–10]. Moreover, machine learning-based finite element methods have also been proposed in the literature [11, 12]. Each particular machine learning approach has its unique error function and corresponding optimization algorithm. Additionally, these techniques usually come with a set of user-defined variables, known as hyperparameters, enabling control over various facets of the behavior of the method.

Scientific Machine Learning (SML) and data science techniques are largely comprised of two general classes of methods [13, 14]. One class of data-driven methods relies on massive data sets and it endeavors to extract the underlying patterns and characteristic features of the system with the goal to first learn the system behavior and train the model [5, 8, 10], and then utilize the trained model for predictive computations. Simultaneously, there is another hybrid approach that is gaining attention lately [6, 7]. It leverages prior knowledge of simplified models that are physically intuitive and interpretable and may also have built-in constraints that are based on known prior information. It then employs high-fidelity data – which is usually sparse – to build frameworks for data-driven discrepancy modeling [15–17]. The objective of SML in this case is to help enhance the accuracy and predictive capability of the surrogate models by embedding high-fidelity data directly in the modeling method.

The advancements in modern sensor technologies have increased the quality of available data for physical systems, thereby creating an opportunity to improve predictive modeling of system dynamics by augmenting the governing first principles with high-fidelity data. An enhanced prediction of the underlying physical processes can be obtained by exploiting the residual between the first-principles theory and the sensor measurements of the dynamical systems. This residual may contain deterministic effects or discrepancies, and dynamics of models with discrepancies can be improved w.r.t. the target system using data-driven approaches. A number of machine learning methods have been proposed to learn model error or discrepancies between a given model and measurements of the system using hybrid data assimilation techniques [15–19]. Through these methods, an improved deterministic model for the physical system is obtained by learning additive correction models for the missing physics. However, these techniques rely on large volumes of high-fidelity data, and significant computational resources are usually needed for learning discrepancy models. Discrepancy modeling for complex nonlinear physical systems has also been proposed using Bayesian calibration [20] and non-intrusive Bayesian state estimation technique [21].

This paper presents a data-driven framework for discrepancy modeling to resolve deterministic model-measurement mismatch. Instead of modeling the system entirely from data, it builds upon the prior knowledge of simplified physics by considering the PDE-based systems as a surrogate model, and then enhances the capability of the model by a novel data embedding technique that assimilates the measured response into the variational structure of the surrogate model [22]. This approach benefits from the sound variational structure of the physics-based modeling methods, while missing physics is compensated for via a loss function that is comprised of a residual term which is based on the difference between a quantity of interest from the physics-based model and the observed value that represents the high-fidelity data. We extend our earlier work on elastostatics [22] to the case of elastodynamics that serves as our model problem. Specifically, we leverage the DDV method to compensate for parameter discrepancy, structure mismatch, and model discrepancy in elastodynamics that constitutes an imperfect model of the system. The proposed method results in a cost-effective solution because the prior partial knowledge of the system behavior that is represented via surrogate models is incorporated into the overall framework to aid the modeling process and to improve prediction accuracy. A more general application of the proposed method can be to complex transient problems involving anisotropy and material nonlinearity [23, 24].

An outline of the paper is as follows: Employing linear elastodynamics as the model problem, Section 2 presents the governing system of equations. Our objective is to embed discrepancy in this model via parameter mismatch, boundary condition inadequacy w.r.t. the target system, as well as model discrepancy in terms of missing physics. Section 3 discusses ideas underlying the proposed method, and the structure of the variational equations is presented. Section 4 presents the mathematical details of the derivation of the method, discusses the time integrator employed, and elaborates on the structure of the loss function. Numerical test cases are presented in Section 5, and Conclusions are drawn in Section 6.

2. Governing Equations for Elastodynamics

Let $\text{Ω}\subset {ℝ}^{n_{\text{sd}}}$ be an open, bounded domain with a piece-wise smooth boundary $\text{Γ}$, where $n_{\text{sd}}\ge 2$ is the number of spatial dimensions. The boundary $\text{Γ}$ is divided into two subsets ${\text{Γ}}_g$ and ${\text{Γ}}_h$ on which Dirichlet and Neumann boundary conditions are applied, respectively. These subsets satisfy ${\text{Γ}}_g\cup {\text{Γ}}_h=\text{Γ}$, ${\text{Γ}}_g\cap {\text{Γ}}_h=\varnothing$.

We consider the displacement form of elasticity as our model problem. The governing equation, boundary and initial conditions, and the constitutive equation are given as follows:

$$\rho \ddot{\mathbf{u}}+\mathbf{\nabla }\cdot \mathbf{\sigma }+\mathbf{b}=\mathbf{0}\text{in}\text{Ω}\times ]0,T[$$ (1)

$$\mathbf{u}=\mathbf{\text{g}}\text{on}{\text{Γ}}_{\text{g}}\times ]0,T[$$ (2)

$$\sigma \cdot \mathbf{n}=\mathbf{h}\text{on}{\text{Γ}}_{\text{h}}\times ]0,T[$$ (3)

$$\sigma =\mathbf{C}\mathit{ϵ}(\mathbf{u})\text{in}\text{Ω}\times ]0,T[$$ (4)

$$\mathbf{u}(\mathbf{\text{x}},0)={\mathbf{u}}_0(\mathbf{\text{x}})\mathbf{x}\in \text{Ω}$$ (5)

$$\dot{\mathbf{u}}(\mathbf{x},0)={\dot{\mathbf{u}}}_0(\mathbf{\text{x}})\mathbf{x}\in \text{Ω}$$ (6)

where $\rho$ is the density, $\mathbf{u}$ is the displacement field, $\dot{\mathbf{u}}$ is velocity, $\ddot{\mathbf{u}}$ is acceleration, $\mathbf{b}$ is the body force, and $\mathbf{\sigma }$ is the Cauchy stress tensor. In the context of linear elasticity $\mathbf{\sigma }=\mathbf{\sigma }(\mathbf{u})=\mathbf{C}\mathit{ϵ}(\mathbf{u})$ wherein $\mathbf{C}$ is the fourth-order tensor of material moduli with major- and minor-symmetries. $\mathit{ϵ}(\mathbf{u})=1/2\left(\mathbf{\nabla }\mathbf{u}+{(\mathbf{\nabla }\mathbf{u})}^T\right)$ is the symmetric part of the displacement gradient, and $\mathbf{n}$ is the outward unit normal. Furthermore, $\mathbf{g}$ is the prescribed displacement at the Dirichlet boundary, $\mathbf{h}$ is the prescribed traction at the Neumann boundary, ${\mathbf{u}}_0$ and ${\dot{\mathbf{u}}}_0$ are the given initial conditions for the displacement and velocity fields, respectively. The functional spaces appropriate for the displacement trial solutions and weighting functions are:

$$\mathcal{S}={\mathcal{S}}_t=\left\{\mathbf{u}(\cdot ,t)\mid \mathbf{u}(\mathbf{x},t)=\mathbf{g}(\mathbf{x},t),\mathbf{x}\in {\text{Γ}}_g,\mathbf{u}(\cdot ,t)\in H^1(\text{Ω})\right\}$$ (7)

$$\mathcal{V}=\left\{\mathbf{w}(\cdot )\mid \mathbf{w}(\mathbf{x})=\mathbf{0},\mathbf{x}\in {\text{Γ}}_g,\mathbf{w}(\cdot )\in H^1(\text{Ω})\right\}$$ (8)

where $L_2(\text{Ω})$ and $H^1(\text{Ω})$ are standard Sobolev spaces of square-integrable functions, and functions with square-integrable first derivatives, respectively. The weak form of the governing equations can be expressed as: Find $\mathbf{u}\in {\mathcal{S}}_t$ such that $\forall \mathbf{w}\in \mathcal{V}$ the following holds.

$$\underset{\text{Ω}}{\int }\mathbf{w}\cdot \rho \ddot{\mathbf{u}}\text{dΩ}+\underset{\text{Ω}}{\int }\nabla \mathbf{w}:\sigma \text{dΩ}=\underset{\text{Ω}}{\int }\mathbf{w}\cdot \mathbf{b}\text{dΩ}+\underset{{\text{Γ}}_h}{\int }\mathbf{w}\cdot \mathbf{h}\text{dΓ}$$ (9)

Let $\mathbf{U}=\{\mathbf{u}(\mathbf{x},t)\}$ and $\mathbf{W}=\{\mathbf{w}(\mathbf{x})\}$. Equation (9) can be written in an abstract form as follows: Find $\mathbf{U}\in {\mathcal{S}}_t$ such that $\forall \mathbf{W}\in \mathcal{V}$.

$$\mathcal{A}(\mathbf{W},\mathbf{U})=\mathcal{L}(\mathbf{W})$$ (10)

where $\mathcal{A}(\mathbf{\text{W}},\mathbf{\text{U}})=\mathcal{M}(\mathbf{\text{W}},\mathbf{\text{U}})+\mathcal{B}(\mathbf{\text{W}},\mathbf{\text{U}})$ are the mass and stiffness matrices that emanate from the terms on the left-hand side of (9), and $\mathcal{L}(\mathbf{W})$ is comprised of the forcing terms on the right-hand side of (9).

3. Discrepancy Modeling

We consider the case where the response of the physical system is known in the form of measured displacement and/or strain field in a subregion (or patch) of the domain $\text{Ω}$. Assuming that the solution of the governing system of equations which is an idealization of the physical system being modeled has a measurable difference from the response of the target system. Accordingly, discrete jumps could appear in the modeled and target values of the field (and/or its gradients) along the discrete patches of dimension $n_d\le n_{sd}$ where sensor measurements are taken. For a 2D problem, this means displacement and/or strain fields that are recorded along discrete lines or on discrete 2D patches can have measurable difference from the predicted values. Therefore, for a numerical method to be able to account for the jump in the measured (target) and modeled (simulated) response at discrete locations in the domain, it should first have the provision to accommodate the jump in the field and its gradients that can then be variationally minimized. Accordingly, we first develop the VMDG form that is endowed with the discontinuity capturing feature [25–28], and then embed in it the measured (target) data via a variationally derived loss function.

3.1. Elastodynamics with Data Embedding Method

We consider the case where displacement data (e.g., sensor-based measured displacements) is available only in a smaller patch ${\text{Ω}}_1$ of a larger domain ${\text{Ω}}_2$. We assume that the model-based predicted values of the displacement field may be different from the measured (target) displacement field in the subdomain ${\text{Ω}}_1$. This means that over the patch ${\text{Ω}}_1$ there can be a finite jump in the solution space between the modeled and the target values. Our objective in the development of the method is to make this jump vanish, thereby making the solution on patch ${\text{Ω}}_1$ converge to the target value. Enforcing this condition via Lagrange multiplier method can introduce potential discontinuity in the displacement field between ${\text{Ω}}_1$ and ${\text{Ω}}_2$. This can however be addressed via the VMDG method [25–28]. Figure 1 illustrates the idea in which ${\text{Ω}}_1$ represents the local subdomain over which the measured data or the local behavior of the system is known, and it overlies the global domain ${\text{Ω}}_2$. Secondly, both domains are coupled together at the interface ${\text{Γ}}_I$. Accordingly, we consider that patch ${\text{Ω}}_1$ has two types of discontinuities in the field at the boundary ${\text{Γ}}_I$; (i) a discontinuity in the solution space between the modeled and target values, and (ii) a discontinuity across the physical boundary ${\text{Γ}}_I$ between ${\text{Ω}}_1$ and ${\text{Ω}}_2$. The latter condition allows the use of discontinuous functions at the interface ${\text{Γ}}_I$ to approximate the displacement field. The induced discontinuity at ${\text{Γ}}_I$ necessitates a method that enforces the continuity of the field and its flux across ${\text{Γ}}_I$. The key idea is to enforce the convergence of the modeled field to that of the measured response in the smaller patch ${\text{Ω}}_1$, and then couple the modeled field from the two domains at ${\text{Γ}}_I$ using the VMDG framework with the objective to improve the overall accuracy of the solution in the entire domain ${\text{Ω}}_1\cup {\text{Ω}}_2$.

Figure 1:

Data patch coupled with the domain via a DG interface.

We start with a variational method that can accommodate discrete embedded interfaces as well as spatial patches in the domain while satisfying the governing balance laws. As a model problem, we consider a 2D domain containing 1D discrete embedded interfaces where pointwise data about the system behavior is known, or 2D spatial subdomains where distributed spatial data on the response of the system is known a-priori. In the context of elasticity, discrete data means pointwise information on the displacement field, and distributed spatial data means information on local stretching or straining that can be obtained via a patch of strain gauges applied to the 2D subdomain.

For simplicity let us assume that we have measured data ${\mathbf{u}}^{(data)}$ that represents the actual behavior of the system at discrete but otherwise arbitrary location ${\text{Γ}}_I$. Our first objective is to enforce the continuity of the modeled response ${\mathbf{u}}_1$ and the measured data ${\mathbf{u}}^{(data)}$.

$${\mathbf{u}}_1-{\mathbf{u}}^{(data)}=\mathbf{0}{\text{on Γ}}_I\times ]0,T[$$ (11)

Enforcing this continuity in the solution space between ${\mathbf{u}}_1$ and ${\mathbf{u}}^{(data)}$ may induce discontinuity in the spatial solution across the subdomains ${\text{Ω}}_1$ and ${\text{Ω}}_2$ at ${\text{Γ}}_I$ thereby necessitating a method that enforces the continuity of the field and its flux in the spatial domain across ${\text{Γ}}_I$. This discontinuity in the displacement field is written as:

$$[\text{}[\mathbf{u}]\text{}]={\text{Π}}_1{\mathbf{u}}_1-{\text{Π}}_2{\mathbf{u}}_2=\mathbf{0}\text{on}{\text{Γ}}_I\times ]0,T[$$ (12)

where ${\text{Π}}_{\alpha }:H^1\left({\text{Ω}}_{\alpha }\right)\to H^{\frac{1}{2}}\left({\text{Γ}}_{\alpha }\right)$ is the trace operator that extends the primary field from the interior to the boundary. In the present work ${\text{Π}}_{\alpha }$ is considered identity.

4. Data-Driven Variational Method (DDV)

Standard modeling methods for physical systems are mathematical idealizations that – subject to the initial and boundary conditions – predict the response of the system where it is subjected to some actions. There are instances when the local behavior of the physical system is known, either through local sensor data or via other quantifiable measurement techniques. These measurements are subsequently employed to assess the quality of the computed solution. In case of deviation of the predicted response from the measured data, the problem is rerun by making adjustments to the model parameters. Many a times parameter adjustments are not enough as the mathematical model may itself be deficient in terms of the physics being modeled. So the question arises, “Is it possible to develop a method where domain interior information on the behavior of the system can be directly utilized in driving the overall system by embedding it in forward calculations?”

In the following, we present a formulation that accommodates embedding of data or the known information on system behavior in a variationally consistent fashion. The formulation is given by:

$${\mathcal{A}}^{\text{Gal}}(\mathbf{W},\mathbf{U})+{\mathcal{B}}_{\text{DG}}^{\text{Interface}}(\mathbf{W},\mathbf{U})+{\mathcal{B}}_{\text{Data}}^{\text{Interface}}(\mathbf{W},\mathbf{U})={\mathcal{L}}^{\text{Gal}}(\mathbf{W})$$ (13)

The explicit expressions for the bilinear and linear forms are defined as follows:

$${\mathcal{A}}^{\text{Gal}}=\sum _{\alpha =1,2}\left[{\left({\mathbf{w}}_{\alpha },{\rho }_{\alpha }{\ddot{\mathbf{u}}}_{\alpha }\right)}_{{\text{Ω}}_{\alpha }}+{\left(ϵ\left({\mathbf{w}}_{\alpha }\right),\sigma \left({\mathbf{u}}_{\alpha }\right)\right)}_{{\text{Ω}}_{\alpha }}\right]$$ (14)

$${\mathcal{B}}_{\text{DG}}^{\text{Interface}}=-{\langle [\text{}[\mathbf{w}]\text{}],\{\sigma (\mathbf{u})\mathbf{n}\}\rangle }_{{\text{Γ}}_I}-{\langle \{\sigma (\mathbf{w})\mathbf{n}\},[\text{}[\mathbf{u}]\text{}]\rangle }_{{\text{Γ}}_I}+{\langle [\text{}[\mathbf{w}]\text{}],{\tau }^D[\text{}[\mathbf{u}]\text{}\rangle }_{{\text{Γ}}_I}$$ (15)

$${\mathcal{B}}_{\text{Data}}^{\text{Interface}}=-{\langle \text{}[[\mathbf{w}]\text{}],{\tau }^D\left({\mathbf{u}}_1-{\mathbf{u}}^{(data)}\right)\rangle }_{{\text{Γ}}_I}+{\langle {\mathbf{w}}_1,{\delta }^D\left({\mathbf{u}}_1-{\mathbf{u}}^{(data)}\right)\rangle }_{{\text{Γ}}_I}-{\langle {\mathbf{w}}_1,{\tau }^D[\text{}[\mathbf{u}]\text{}]\rangle }_{{\text{Γ}}_I}$$ (16)

$${\mathcal{L}}^{\text{Gal}}=\sum _{\alpha =1,2}\left[{\left({\mathbf{w}}_{\alpha },{\mathbf{b}}_{\alpha }\right)}_{{\text{Ω}}_{\alpha }}+{\left({\mathbf{w}}_{\alpha },{\mathbf{h}}_{\alpha }\right)}_{{\text{Γ}}_{\alpha }}\right]$$ (17)

where $(\cdot ,\cdot )=\int (\cdot )d\text{Ω}$ is the $L_2$ inner product, and ${\langle \cdot ,\cdot \rangle }_{{\text{Γ}}_I}$ is the duality pairing at ${\text{Γ}}_I$. ${\tau }^D$ and ${\delta }^D$ are the interface stabilization tensors that arise because of the consistent derivation presented in the following sections. Furthermore, the variationally consistent derivation yields additional flux-type terms that can be employed to embed data. However, to keep the discussion simple we have suppressed them, and only presented the terms that have been employed in the numerical solution section of the paper.

For numerical implementation, the matrix form of the stiffness terms and the vector form of the force terms are given in equ. (40) and (41) respectively.

4.1. Derivation of Data-Driven Method

We start with the Lagrange multiplier method that enforces the continuity of modeled and measured response in ${\text{Ω}}_1$ as well as the continuity of the displacement field and balance of tractions between ${\text{Ω}}_1$ and ${\text{Ω}}_2$ at ${\text{Γ}}_I$.

The weak form of the problem is constructed by taking the inner product of vector-valued test functions $\mathbf{w}$ with the balance of linear momentum equation and integrating over the domain, as well as taking the inner product of another vector-valued test function $\mu$ with the continuity equation and integrating along the common interface ${\text{Γ}}_I$ between ${\text{Ω}}_1$ and ${\text{Ω}}_2$. In addition to the spaces of functions $\mathbf{u}$ and $\mathbf{w}$ specified earlier, we specify the space of functions for $\mu$ as

$$\mathcal{Q}=\left\{\mathit{\mu }(x)\mid \mathit{\mu }\in H^{-1/2}\left({\text{Γ}}_I\right)\right\}$$ (18)

After applying integration by parts, the weak form of the problem can be stated as follows: Given ${\rho }_{\alpha }$, ${\mathbf{b}}_{\alpha }$, ${\mathbf{g}}_{\alpha }$, ${\mathbf{h}}_{\alpha }$, ${\left({\mathbf{u}}_{\alpha }\right)}_0$, and ${\left({\dot{\mathbf{u}}}_{\alpha }\right)}_0$; find ${\mathbf{u}}_{\alpha }\in {\mathcal{S}}_t^{\alpha }$ and $\mathbf{\lambda }\in \mathcal{Q}$ such that $\forall {\mathbf{w}}_{\alpha }\in {\mathcal{V}}^{\alpha }$ and $\forall \mathit{\mu }\in \mathcal{Q}$ for $\alpha =1,2$, the following holds:

$$\begin{gathered} \mathcal{A}{\left({\mathbf{W}}_1,{\mathbf{U}}_1\right)}_{{\text{Ω}}_1}+\mathcal{A}{\left({\mathbf{W}}_2,{\mathbf{U}}_2\right)}_{{\text{Ω}}_2}+{\langle [\text{}[\mathbf{w}]\text{}],\mathit{\lambda }\rangle }_{{\text{Γ}}_I}+{\langle \mathit{\mu },[\text{}[\mathbf{u}]\text{}]\rangle }_{{\text{Γ}}_I} \\ +{\langle {\mathbf{w}}_1,{\mathit{\lambda }}_D\rangle }_{{\text{Γ}}_I}+{\langle {\mathit{\mu }}_D,\left({\mathbf{u}}_1-{\mathbf{u}}^{(data)}\right)\rangle }_{{\text{Γ}}_I}=\mathcal{L}{\left({\mathbf{W}}_1\right)}_{{\text{Ω}}_1}+\mathcal{L}{\left({\mathbf{W}}_2\right)}_{{\text{Ω}}_2} \end{gathered}$$ (19)

In equ. (19) the first two terms enforce balance laws in the subdomain interiors, the terms ${\langle [\text{}[\mathbf{w}]\text{}],\mathit{\lambda }\rangle }_{{\text{Γ}}_I}$ and ${\langle \mathit{\mu },[\text{}[\mathbf{u}]\text{}]\rangle }_{{\text{Γ}}_I}$ impose balance of tractions and continuity of the field across the embedded interface ${\text{Γ}}_I$ using Lagrange multipliers $\mathit{\lambda }$ and $\mathit{\mu }$, respectively. In addition, the terms ${\langle {\mathbf{w}}_1,{\mathit{\lambda }}_D\rangle }_{{\text{Γ}}_I}$ and ${\langle {\mathit{\mu }}_D,\left({\mathbf{u}}_1-{\mathbf{u}}^{(data)}\right)\rangle }_{{\text{Γ}}_I}$ enforce data constraint at the boundary of the subdomain ${\text{Ω}}_1$ via Lagrange multipliers ${\mathit{\lambda }}_D$ and ${\mathit{\mu }}_D$.

Instead of numerically solving for the interfacial Lagrange multiplier field, our objective is to derive analytical expressions for $\mathit{\lambda }$ and ${\mathit{\lambda }}_D$ in terms of the underlying displacement field at the interface ${\text{Γ}}_I$.

We now derive the semi-discrete form (discrete in space, continuous in time) of the problem, and employ the VMDG method to recover a primal field formulation with consistently derived transmission conditions for interface coupling of the solution. We assume that along the interface ${\text{Γ}}_I$ where we are weakly imposing the continuity of the displacement field, there exists a small error as compared to the case where full continuity of the field was enforced via the use of continuous functions. We term this error as the fine-scale field. We now outline the key steps of the derivation: First, the primary field (displacements) and its associated test function will be decomposed into coarse- and fine-scale contributions. A likewise decomposition of the weighting function into coarse- and fine-scale fields, and using the linearity of the weighting function slot, gives rise to coarse- and fine-scale weak subproblems of the balance of linear momentum equation. By expanding the fine-scale field in terms of bubble functions [26], and with the appropriate assumptions, the fine-scale problem is solved for the fine-scale field in terms of the coarse scales and Lagrange multiplier $\mathit{\lambda }$. Then, this solution is substituted in the weak form of the continuity equation. Employing arguments about the stability of the problem and using the properties of the function space of $\mathit{\lambda }$ and $\mathit{\mu }$, the solution for $\mathit{\lambda }$ is written in terms of the coarse-scale displacement field. Finally, both the fine displacement field and $\mathit{\lambda }$ are substituted in the coarse-scale weak form of the balance of linear momentum equation, in which the only unknown that remains to be solved is the coarse-scale displacement field. This modified coarse-scale problem also includes interface terms containing parameters that capture the effect of the fine scales and the traction at the interface, thereby providing the coupling between the subdomains ${\text{Ω}}_1$ and ${\text{Ω}}_2$.

4.2. Multiscale Decomposition of the Displacement Field

In [25–28] we presented the Variational Multiscale Discontinuous Galerkin (VMDG) method that derives from the hierarchical structure of the VMS framework [29–31] wherein the interface coupling terms were derived via local split of the displacement field in a narrow band along the interface ${\text{Γ}}_I$ into coarse and fine-scale parts.

$${\mathbf{u}}_{\alpha }(\mathbf{x},t)=\underset{\text{coarse scale}}{\underbrace{{\overline{\mathbf{u}}}_{\alpha }(\mathbf{x},t)}}+\underset{\text{fine scale}}{\underbrace{{\mathbf{u}}_{\alpha }^{\prime }(\mathbf{x},t)}}$$ (20)

$${\mathbf{w}}_{\alpha }(\mathbf{x})={\overline{\mathbf{w}}}_{\alpha }(\mathbf{x})+{\mathbf{w}}_{\alpha }^{\prime }(\mathbf{x})$$ (21)

Differentiating the displacement field in time leads to coarse and fine-scale velocity and acceleration fields. In order to simplify the formulation, we assume that the effect of the time variation of the fine-scale field is negligible, so that

$${\dot{\mathbf{u}}}_{\alpha }\approx {\dot{\overline{\mathbf{u}}}}_{\alpha },{\ddot{\mathbf{u}}}_{\alpha }\approx {\ddot{\overline{\mathbf{u}}}}_{\alpha }$$ (22)

Employing equ. (20) and (21), together with equ. (22), and using VMS ideas results in two hierarchically connected variational equations.

Coarse-scale problem

$$\begin{gathered} \mathcal{A}{\left({\overline{\mathbf{W}}}_1,\left({\overline{\mathbf{U}}}_1+{\mathbf{U}}_1^{\prime }\right)\right)}_{{\text{Ω}}_1}+\mathcal{B}{\left({\overline{\mathbf{W}}}_2,\left({\overline{\mathbf{U}}}_2+{\mathbf{U}}_2^{\prime }\right)\right)}_{{\text{Ω}}_2}+{\langle [\text{}[\overline{\mathbf{w}}]\text{}],\mathit{\lambda }\rangle }_{{\text{Γ}}_I}+{\langle \mathit{\mu },[\text{}[\overline{\mathbf{u}}+{\mathbf{u}}^{\prime }]\text{}]\rangle }_{{\text{Γ}}_I} \\ +{\langle {\overline{\mathbf{w}}}_1,{\mathit{\lambda }}_D\rangle }_{{\text{Γ}}_I}+{\langle {\mathit{\mu }}_D,\left({\overline{\mathbf{u}}}_1+{\mathbf{u}}_1^{\prime }-{\mathbf{u}}^{(data)}\right)\rangle }_{{\text{Γ}}_I}=\mathcal{L}{\left({\overline{\mathbf{W}}}_1\right)}_{{\text{Ω}}_1}+\mathcal{L}{\left({\overline{\mathbf{W}}}_2\right)}_{{\text{Ω}}_2} \end{gathered}$$ (23)

Fine-scale problem

$$\begin{gathered} \mathcal{B}{\left({\mathbf{W}}_1^{\prime },\left({\overline{\mathbf{U}}}_1+{\mathbf{U}}_1^{\prime }\right)\right)}_{{\text{Ω}}_1}+\mathcal{B}{\left({\mathbf{W}}_2^{\prime },\left({\overline{\mathbf{U}}}_2+{\mathbf{U}}_2^{\prime }\right)\right)}_{{\text{Ω}}_2}+{\langle [\text{}[{\mathbf{w}}^{\prime }]\text{}],\mathit{\lambda }\rangle }_{{\text{Γ}}_I} \\ +{\langle {\mathbf{w}}_1^{\prime },{\mathit{\lambda }}_D\rangle }_{{\text{Γ}}_I}=\mathcal{L}{\left({\mathbf{W}}_1^{\prime }\right)}_{{\text{Ω}}_1}+\mathcal{L}{\left({\mathbf{W}}_2^{\prime }\right)}_{{\text{Ω}}_2} \end{gathered}$$ (24)

In equ. (23) and (24), the unknown fields are ${\overline{\mathbf{u}}}_{\alpha }$, ${\mathbf{u}}_{\alpha }^{\prime }$, $\mathit{\lambda }$, and ${\mathit{\lambda }}_D$. Our objective is to locally resolve equ. (24) to extract analytical expressions for ${\mathbf{u}}_{\alpha }^{\prime }$, $\mathit{\lambda }$, and ${\mathit{\lambda }}_D$ that can be substituted in equ. (23). The resulting form will then be a function of ${\overline{\mathbf{u}}}_{\alpha }$ which will be solved numerically.

Accordingly, the fine scale problem is localized to a narrow band across the interface ${\text{Γ}}_I$. In the discrete setting this can be viewed as the first layer of elements across ${\text{Γ}}_I$, as shown in Fig. 2. ${\gamma }_D$ is a measure of the interface between two elements along ${\text{Γ}}_I$.

Figure 2:

Interface segment ${\gamma }_D$.

Remark 1.

In order to keep the presentation of ideas simple, the inertial effects are assumed to be zero at the fine scale level.

Remark 2.

Multiscale decomposition is only applied to the displacement field. This is sufficient to enforce the continuity of the fields and simplifies subsequent derivation of the Lagrange multipliers.

4.3. Interfacial Fine-Scale Problem

We write equ. (24) in a residual form as follows:

$${\mathcal{B}}_{\alpha }{\left({\mathbf{W}}_{\alpha }^{\prime },{\mathbf{U}}_{\alpha }^{\prime }\right)}_{{\text{Ω}}_{\alpha }}=-{\langle {\mathbf{w}}_1^{\prime },\mathit{\lambda }+{\mathit{\lambda }}_D\rangle }_{{\text{Γ}}_I}+{\langle {\mathbf{w}}_2^{\prime },\mathit{\lambda }\rangle }_{{\text{Γ}}_I}+\sum _{\alpha =1,2}\left[{\mathcal{L}}_{\alpha }{\left({\mathbf{W}}_{\alpha }^{\prime }\right)}_{{\text{Ω}}_{\alpha }}-{\mathcal{B}}_{\alpha }{\left({\mathbf{W}}_{\alpha }^{\prime },{\overline{\mathbf{U}}}_{\alpha }\right)}_{{\text{Ω}}_{\alpha }}\right]$$ (25)

We then resolve equ. (25) under some reasonable assumptions: (i) fine scales are assumed localized to the interface, and (ii) fine scales are a combination of edge bubble functions [31].

$${\mathbf{u}}_{\alpha }^{\prime }{|}_{{\omega }_{\alpha }^D}={\mathit{\beta }}_{\alpha }{b}_{\alpha }(\mathbf{\text{x}})$$ (26)

$${\mathbf{w}}_{\alpha }^{\prime }{|}_{{\omega }_{\alpha }^D}={\mathit{\eta }}_{\alpha }{b}_{\alpha }(\mathbf{\text{x}})$$ (27)

Expanding the fine scale fields in terms of edge functions and locally resolving the variational equation (25) segment-wise we arrive at the analytical expression of the fine scale displacement fields across ${\text{Γ}}_I$ in domain ${\text{Ω}}_{\alpha }(\alpha =1,2)$.

$${\mathbf{u}}_{\alpha }^{\prime }{|}_{{\gamma }_D}={\mathit{\tau }}_{\alpha }^D\left[{(-1)}^{\alpha }\mathit{\lambda }+(\alpha -2){\mathit{\lambda }}_D-{\mathbf{n}}_{\alpha }\cdot {\mathit{\sigma }}_{\alpha }\left({\mathbf{u}}_{\alpha }^{\prime }\right)\right]$$ (28)

It is important to note that ${\mathbf{u}}_{\alpha }^{\prime }{|}_{{\gamma }_D}$ is proportional to the residual of boundary fluxes. In ${\text{Ω}}_1$, there is another flux term that is based on the potential jump between solution ${\mathbf{u}}_1^h$ in ${\text{Ω}}_1$, and the target value of ${\mathbf{u}}^{(data)}$. The stabilization tensor ${\mathit{\tau }}_{\alpha }^D$ is given by:

$${\mathit{\tau }}_{\alpha }^D={\left[\text{meas}\left({\gamma }_D\right)\right]}^{-1}{\left(\underset{{\gamma }_D}{\int }{b}_{\alpha }\text{dΓ}\right)}^2{\left[{\left(ϵ\left(b_{\alpha }{\mathbf{e}}_i\right),\mathit{\sigma }\left(b_{\alpha }{\mathbf{e}}_j\right)\right)}_{{\text{Ω}}_{\alpha }}{\mathbf{e}}_i\otimes {\mathbf{e}}_j\right]}^{-1}$$ (29)

However, the expression (28) has Lagrange multipliers in it, which are still the unknown fields. To derive expressions that are a function of the displacement field alone, we consider (i) the coarse-scale continuity equ. (12) that is comprised of a jump in the displacement field, which in turn is a function of coarse and fine-scale fields, and (ii) a continuity equation that enforces data constraint (11):

In the physical domain:

$${\langle \mathit{\mu },[\text{}[\overline{\mathbf{u}}]\text{}]\rangle }_{{\text{Γ}}_I}+{\langle \mathit{\mu },{\mathit{\tau }}_1^D\left[-\mathit{\lambda }-{\mathit{\lambda }}_D-{\overline{\sigma }}_1{\mathbf{n}}_1\right]\rangle }_{{\text{Γ}}_I}+{\langle \mathit{\mu },{\mathit{\tau }}_2^D\left[-\mathit{\lambda }+{\overline{\sigma }}_2{\mathbf{n}}_2\right]\rangle }_{{\text{Γ}}_I}=0$$ (30)

In the solution space:

$${\langle {\mathit{\mu }}_D,\left({\overline{\mathbf{u}}}_1-{\mathbf{u}}^{(data)}\right)\rangle }_{{\text{Γ}}_I}+{\langle {\mathit{\mu }}_D,{\mathit{\tau }}_1^D\left[-\mathit{\lambda }-{\mathit{\lambda }}_D-{\overline{\mathit{\sigma }}}_1{\mathbf{n}}_1\right]\rangle }_{{\text{Γ}}_I}=0$$ (31)

Employing piecewise $L_2$ projection of $\mathit{\lambda }$ and ${\mathit{\lambda }}_D$, we obtain segment-wise discontinuous multiplier (numerical flux) as follows:

$$\mathit{\lambda }{|}_{{\gamma }_D}={\mathit{\tau }}_2^D([\text{}[\overline{\mathbf{u}}]\text{}]){|}_{{\gamma }_D}+\left({\overline{\mathit{\sigma }}}_2{\mathbf{n}}_2\right){|}_{{\gamma }_D}-{\mathit{\tau }}_2^D\left({\overline{\mathbf{u}}}_1-{\mathbf{u}}^{(data)}\right){|}_{{\gamma }_D}$$ (32)

$${\mathit{\lambda }}_D{|}_{{\gamma }_D}={\mathit{\tau }}_2^D([\text{}[\overline{\mathbf{u}}]\text{}]){|}_{{\gamma }_D}-\left({\overline{\mathit{\sigma }}}_1{\mathbf{n}}_1+{\overline{\mathit{\sigma }}}_2{\mathbf{n}}_2\right){|}_{{\gamma }_D}+\left({\mathit{\tau }}_1^D+{\mathit{\tau }}_2^D\right)\left({\overline{\mathbf{u}}}_1-{\mathbf{u}}^{(data)}\right){|}_{{\gamma }_D}$$ (33)

Embedding $\left[\mathit{\lambda },{\mathit{\lambda }}_D,{\mathbf{u}}^{\prime }\right]$ in coarse-scale formulation (23) leads to the formulation where data is embedded via a variationally consistent procedure termed as the Data-Driven Variational (DDV) method.

(34)

where ${\mathit{\tau }}^D$ and ${\mathit{\delta }}^D$ are the interface stabilization tensors given as follows:

$${\mathit{\tau }}^D={\left({\mathit{\tau }}_2^D\right)}^{-1}$$

$${\mathit{\delta }}^D=\left[{\left({\mathit{\tau }}_1^D\right)}^{-1}+{\left({\mathit{\tau }}_2^D\right)}^{-1}\right]$$

and $\{\mathbf{\sigma }(\mathbf{u})\mathbf{n}\}$ is the average stress at the interface.

$$\{\sigma (\mathbf{u})\mathbf{n}\}={\sigma }_1\left({\mathbf{u}}_1\right){\mathbf{n}}_1-{\sigma }_2\left({\mathbf{u}}_2\right){\mathbf{n}}_2$$

We want to highlight that in equ. (34) the DG terms possess the features of Nitsche-type methods. However, by employing the present approach, weighting parameters can be consistently derived. DDV formulation can then be written in an abstract form. Let $\mathbf{U}=\{\mathbf{u}(\mathbf{x},t)\}$ and $\mathbf{W}=\{\mathbf{w}(\mathbf{x})\}$. Find $\mathbf{U}\in {\mathcal{S}}_t$ such that $\forall \mathbf{W}\in \mathcal{V}$.

$${\mathcal{A}}^{\text{Gal}}(\mathbf{W},\mathbf{U})+{\mathcal{B}}_{\text{DG}}^{\text{Interface}}(\mathbf{W},\mathbf{U})+{\mathcal{B}}_{\text{Data}}^{\text{Interface}}(\mathbf{W},\mathbf{U})={\mathcal{L}}^{\text{Gal}}(\mathbf{W},\mathbf{U})$$ (35)

The explicit expressions for the bilinear and linear forms are defined as follows:

$${\mathcal{A}}^{\text{Gal}}=\sum _{\alpha =1,2}\left[{\left({\mathbf{w}}_{\alpha },{\rho }_{\alpha }{\ddot{\mathbf{u}}}_{\alpha }\right)}_{{\text{Ω}}_{\alpha }}+{\left(\mathit{ϵ}\left({\mathbf{w}}_{\alpha }\right),\mathit{\sigma }\left({\mathbf{u}}_{\alpha }\right)\right)}_{{\text{Ω}}_{\alpha }}\right]$$ (36)

$${\mathcal{B}}_{\text{DG}}^{\text{Interface}}=-{\langle [\text{}[\mathbf{w}]\text{}],\{\mathit{\sigma }(\mathbf{u})\mathbf{n}\}\rangle }_{{\text{Γ}}_I}-{\langle \{\mathit{\sigma }(\mathbf{w})\mathbf{n}\},[\text{}[\mathbf{u}]\text{}\rangle }_{{\text{Γ}}_I}+\langle [\text{}[\mathbf{w}]\text{}],{\tau }^D{[\text{}[\mathbf{u}]\text{}]\rangle }_{{\text{Γ}}_I}$$ (37)

$$\begin{gathered} {\mathcal{B}}_{\text{Data}}^{\text{Interface}}=-\langle [\text{}[\mathbf{w}]\text{}],{\tau }^D{\left({\mathbf{u}}_1-{\mathbf{u}}^{(data)}\right)\rangle }_{{\text{Γ}}_I}+\langle {\mathbf{w}}_1,{\delta }^D{\left({\mathbf{u}}_1-{\mathbf{u}}^{(data)}\right)\rangle }_{{\text{Γ}}_I} \\ -\langle {\mathbf{w}}_1,{\tau }^D{[\text{}[\mathbf{u}]\text{}]\rangle }_{{\text{Γ}}_I}-\langle {\mathbf{w}}_1,{\sigma }_2\left({\mathbf{u}}_2\right){\mathbf{n}}_2{\rangle }_{{\text{Γ}}_I}-\langle {\mathbf{w}}_2,{\sigma }_1\left({\mathbf{u}}_1\right){\mathbf{n}}_1{\rangle }_{{\text{Γ}}_I} \\ -\langle {\sigma }_1\left({\mathbf{w}}_1\right){\mathbf{n}}_1,{\mathbf{u}}_2-{\mathbf{u}}^{(data)}{\rangle }_{{\text{Γ}}_I}-\langle {\sigma }_2\left({\mathbf{w}}_2\right){\mathbf{n}}_2,{\mathbf{u}}_1-{\mathbf{u}}^{(data)}{\rangle }_{{\text{Γ}}_I} \end{gathered}$$ (38)

$${\mathcal{L}}^{\text{Gal}}=\sum _{\alpha =1,2}\left[{\left({\mathbf{w}}_{\alpha },{\mathbf{b}}_{\alpha }\right)}_{{\text{Ω}}_{\alpha }}+{\left({\mathbf{w}}_{\alpha },{\mathbf{h}}_{\alpha }\right)}_{{\text{Γ}}_{\alpha }}\right]$$ (39)

To keep the method simple, in the numerical implementation of the method we retain the displacement jump terms for the embedding of the data, while we drop the flux jump terms, as shown in equ. (16).

Remark 3.

Appendix A presents the implementation of the DDV formulation in the context of the finite element method.

4.4. Time Discretization

In developing numerical solutions to the system of equations in (35), the time interval of interest $\left]0,T\right[$ is partitioned into $N$ sub-intervals $\left(t_n,t_{n+1}\right)$, with $t_n<t_{n+1}$, $n\in \mathbb{N}$, $n<N$. In this work, we consider all time subintervals to be of equal size $t_{n+1}-t_n=\text{Δ}t=T/N$. Since the measured data is assumed given in the form of displacement field – i.e., displacement time history – and that the data embedding terms are also written in terms of the displacement field, we write the Newmark predictor-corrector method in the displacement form as given in Algorithm 1.

Remark 4.

We have intentionally employed a simple time integration scheme so that we can highlight the salient mathematical attributes of the proposed method that improve the accuracy and stability of the formulation.

Algorithm 1:

Displacement form of the Newmark predictor-corrector method.

where

$$\mathbf{K}=\sum _{\alpha =1,2}\left[\underset{\text{Ω}}{\int }\nabla {\mathbf{w}}_{\alpha }\left({\mathbf{C}}_{\alpha }ϵ\left({\mathbf{u}}_{\alpha }\right)\right)d\text{Ω}+\underset{{\text{Γ}}_I}{\int }{\mathbf{w}}_{\alpha }\left({\tau }_{\alpha }^D{\mathbf{u}}_{\alpha }\right)d\text{Γ}\right]$$ (40)

$${\mathbf{F}}_{n+1}=\sum _{\alpha =1,2}\left[\underset{\text{Ω}}{\int }{\mathbf{w}}_{\alpha }{\mathbf{b}}_{\alpha }d\text{Ω}+\underset{{\text{Γ}}_h}{\int }{\mathbf{w}}_{\alpha }{h}_{\alpha }d\text{Γ}+\underset{{\text{Γ}}_I}{\int }{\mathbf{w}}_{\alpha }\left({\tau }_{\alpha }^D{\mathbf{u}}^{(data)}\right)d\text{Γ}\right]$$ (41)

4.5. Interpretation of Discrepancy Term as a Loss Term

Linear regression is a common statistical / machine learning tool for modeling the relationship between some input variables and some real-valued outputs using a loss function. A loss function $\mathcal{l}(q(\mathbf{x};\mathbf{w},b),y)$ determines how far is the prediction $q(\mathbf{x};\mathbf{w},b)$ from the target value $y.q(\mathbf{x};\mathbf{w},b)$ is a function parametrized by a weight vector $\mathbf{w}\in {ℝ}^d$, a bias $b\in ℝ$, that takes $\mathbf{x}$ as an input vector and returns the scalar output $q(\mathbf{x};\mathbf{w},b)=\langle \mathbf{w},\mathbf{x}\rangle +b$. One common approach is to use a squared-loss function in regression, defined as:

$$\mathcal{l}(q(\mathbf{x};\mathbf{w},b),y)={(q(\mathbf{x};\mathbf{w},b)-y)}^2$$ (42)

The empirical risk function corresponding to this loss function is called the Mean Squared Error (MSE), which is defined on a given training $S$ with $m$ values as:

$$L_S(q(\mathbf{x};\mathbf{w},b),y)=\frac{1}{m}\sum _{i=1}^m{\left(q\left({\mathbf{x}}_i;\mathbf{w},b\right)-y_i\right)}^2$$ (43)

The least squares method facilitates an algorithm that solves the empirical risk minimization (ERM) problem for linear regression predictors with respect to the squared loss. The ERM problem for least squares is given as:

$$\underset{\mathbf{w},b}{\arg \min }{L}_S(q(\mathbf{x};\mathbf{w},b),y)=\underset{\mathbf{w},b}{\arg \min }\frac{1}{m}\sum _{i=1}^m{\left(q\left({\mathbf{x}}_i;\mathbf{w},b\right)-y_i\right)}^2$$ (44)

The solution to the above problem gives parameters that minimize the error.

In equ. (38) a residual function arises that is the difference between the computed solution and the sensor-based measured data over subdomain ${\text{Ω}}_1$. This residual $\mathbf{r}\left({\mathbf{u}}_1,{\mathbf{u}}^{(data)}\right)=\left({\mathbf{u}}_1^{(\mathit{model})}-{\mathbf{u}}^{(data)}\right)$ defines how far off is the computed solution from the target values over ${\text{Ω}}_1$. After simplifying equ. (38), the residual error at the interface (i) between modeled solution and measured data in ${\text{Ω}}_1$, and (ii) the jump in the solution between ${\text{Ω}}_1$ and ${\text{Ω}}_2$ across ${\text{Γ}}_I$, is ${\langle {\mathbf{w}}_{\alpha },{\mathit{\tau }}_{\alpha }^D\left({\mathbf{u}}_{\alpha }-{\mathbf{u}}^{(data)}\right)\rangle }_{{\text{Γ}}_I}$.

We make a crucial observation that data terms in ${\langle {\mathbf{w}}_{\alpha },{\tau }_{\alpha }^D\left({\mathbf{u}}_{\alpha }-{\mathbf{u}}^{(data)}\right)\rangle }_{{\text{Γ}}_I}$ behave like a least-squares type linear regression function with the corresponding empirical risk minimization (ERM) problem given by:

$$\begin{gathered} \underset{{\mathbf{u}}_{\alpha }}{\arg \min }{L}_S\left(l\left({\mathbf{u}}_{\alpha },{\mathbf{u}}^{(data)}\right),{\tau }_{\alpha }^D\right)=\underset{{\mathbf{u}}_{\alpha }}{\arg \min }\frac{1}{2}\langle {\tau }_{\alpha }^D{\mathbf{r}}_{\alpha }{\left({\mathbf{u}}_{\alpha }-{\mathbf{u}}^{(data)}\right)\rangle }_{{\text{Γ}}_I}^2 \\ =\underset{{\mathbf{u}}_{\alpha }}{\arg \min }\frac{1}{2}\langle {\tau }_{\alpha }^D{\left({\mathbf{u}}_{\alpha }-{\mathbf{u}}^{(data)}\right)\rangle }_{{\text{Γ}}_I}^2 \end{gathered}$$ (45)

where $l\left(\mathbf{u},{\mathbf{u}}^{(data)}\right)\equiv {\mathbf{r}}_{\alpha }\left({\mathbf{u}}_{\alpha }-{\mathbf{u}}^{(data)}\right)=\left({\mathbf{u}}_{\alpha }-{\mathbf{u}}^{(data)}\right)$ is the squared loss function.

Remark 5.

The advantage of embedding data through the residual is that it retains the variational consistency of the formulation while improving the accuracy of the solution.

4.6. Interpretation of the Stabilization Tensor ${\mathit{\tau }}^D$ as a Kernel Function

In the VMS method, a solution field $\mathbf{u}$ is decomposed into coarse scales $\overline{\mathbf{u}}$, which can be resolved by a given mesh, and unresolved or missing scales ${\mathbf{u}}^{\prime }$ that can be considered as the error. In other words, ${\mathbf{u}}^{\prime }$ represents the orthogonal component of the solution to the PDE that lies in a higher dimensional space as shown in Fig. 3.

Figure 3:

Decomposition of the solution field into coarse and fine scale components.

As described in Section 4.3, fine scales are resolved as a function of the residual of coarse scale equations at ${\text{Γ}}_I$, pre-multiplied and scaled by the tensor ${\mathit{\tau }}^D$. In fact, tensor ${\mathit{\tau }}^D$ projects a low-dimensional component of the solution, i.e., the residual of the coarse scales, onto the fine scales that are the high-dimensional (i.e., higher resolution) components of the solution. Therefore, ${\mathit{\tau }}^D$ which also has error information embedded in it, plays the role of a kernel function [2].

Since a kernel function, through a mapping, projects a lower dimensional space to a richer or higher dimensional space, it can be used to incorporate a-priori knowledge about the response of the system at hand. From the structure of ${\mathit{\tau }}^D$ given in equ. (29), we see that it is symmetric and positive definite, which are the essential properties of a kernel function or its associated Gram matrix. Thus, in equ. (38) and (45) when ${\mathit{\tau }}^D$ acts on the loss function, it magnifies its impact by enriching its space and we see a significant improvement in the accuracy of the solution field even when the data is embedded in a very small patch of the domain.

5. Numerical Experiments

In this section, we present numerical test cases in two dimensions to showcase the variational consistency, spatial and temporal accuracy, and numerical stability of the DDV method for problems in elastodynamics. We have employed the standard Galerkin formulation in space and the average acceleration method in time, which is a member of the Newmark family of predictor-corrector methods with $\beta =1/2$ and $\gamma =1/4$ [32]. We have intentionally employed this simple time integrator so that we can highlight the attributes of amplitude and phase error correction of the DDV method. However, we want to point out that other time integrators of choice can also be employed, e.g., the HHT-$\alpha$ method. Sufficiently high quadrature rules are employed to evaluate all the integrals accurately.

5.1. Convergence Study: L-shaped Domain

To validate the ideas presented in Section 4, we consider an L-shaped domain that is modeled with linear elasticity. The body is loaded such that it produces a state of deformation corresponding to Mode-1 fracture. The exact solution derived from the theory of elasticity is given in equ. (46) – (47) and exhibits stress singularity at the re-entrant corner [33, 34]. Since an exact solution exists, this problem serves as a mathematical test bed for the convergence rate study. From finite element theory, the convergence rate for this class of problems is governed by the regularity of the solution [33, 35]. For global mesh refinement and without the use of singular functions around the singularity, the rate of convergence in the $H^1$-seminorm is limited to $\lambda =0.545$. The geometric description of the problem is shown in Fig. 4. Tractions, that are derived from the exact solution, are applied at all the edges of the domain. The mesh is appropriately constrained to remove the rigid body modes and plane strain conditions are enforced.

Figure 4:

L-shaped domain: (a) problem description, and (b) boundary conditions.

$$u_x^{\text{exact}}=\frac{r^{\lambda }}{2G}[\{k-Q(\lambda +1)\}\cos (\lambda \theta )-\lambda \cos ((\lambda -2)\theta )]$$ (46)

$$u_y^{\text{exact}}=\frac{r^{\lambda }}{2G}[\{k+Q(\lambda +1)\}\sin (\lambda \theta )+\lambda \sin ((\lambda -2)\theta )]$$ (47)

The following parameters have been used in this study:

$$\begin{gathered} \lambda =0.54448373678246398,Q=0.54307557883673652 \\ E=7.5\times {10}^7,G=\frac{E}{2(1+\nu )},k=3-4\nu \end{gathered}$$ (48)

To make this numerical study all-inclusive, we embed in this model problem features that are typically encountered in problems of engineering interest. We start with compatible meshes of 4-node quadrilaterals for convergence rate study of the non-smooth solution because of stress singularity at the re-entrant corner (Section 5.1.1)). We then extend the numerical study to include non-matching spatial meshes (Section 5.1.2)). Accordingly, we divide the domain into two parts with a DG interface inserted as shown by the dashed line in Figure 4b. We further employ two subcases to analyze the effect of embedding high-fidelity data on enhancing the spatial accuracy of the computed solution. Three different spatial locations for two types of data are considered i.e., (i) distributed data in ${ℝ}^{n_{\text{sd}}}$ over a patch, and (ii) discrete data in ${ℝ}^{n_{\text{sd}}-1}$. As shown in Fig. 5, locations 1 and 2 have spatially distributed displacements over a patch (in a continuous form) while location 3 has a discrete displacement field along the edge of the patch.

Figure 5:

Conforming mesh of Q4 quadrilateral elements with embedded data in discrete and continuous form.

5.1.1. L-shaped domain with conforming meshes

The first test case is an L-shaped domain that is modeled using conforming meshes and is governed by the displacement form of elasticity presented in the previous section. Poisson’s ratio is set equal to 0.3. The problem is run using the DDV method presented in Section 2, and spatial accuracy as well as mathematical convergence properties of the method are evaluated.

Table 1 provides details of the mesh hierarchy which is used to carry out the convergence rate study. For both cases, convergence rates in the $L_2$ norm and $H^1$ seminorm of error in the displacement field are shown in Fig. 6. The convergence rate in $H^1$ seminorm for both cases is close to 0.545 as predicted by the finite element theory [33, 35].

Table 1:

Description of the Q4 conforming meshes used in the convergence study.

Mesh	Quadrilateral (Q4)
	Elements	Nodes
Mesh 1	48	70
Mesh 2	192	234
Mesh 3	768	850
Mesh 4	3072	3234

Figure 6:

Convergence rates of standard displacement error for conforming Q4 meshes.

Moreover, it is observed that although the type of data (discrete versus distributed) and the spatial location where the measured data is embedded may be different, all these data-embedding variations uniformly result in a reduced absolute error. The reduction in error is more significant in the $L_2$ norm as compared to the $H^1$ seminorm.

5.1.2. L-shaped domain with non-conforming meshes

In this test case, the L-shaped domain is comprised of non-conforming meshes with a DG interface as shown in Fig. 7. Problem description and boundary conditions are the same as given in Fig. 4. The exact solution derived from the theory of elasticity [33, 34] is given in equ. (46) – (47) with parameters given in equ. (48). Poisson’s ratio is 0.3.

Figure 7:

Non-conforming mesh of 4-node quadrilaterals with embedded data in discrete and continuous form.

Table 2 provides details of the hierarchical mesh refinement that is employed to carry out the convergence rate study. Figure 8 presents convergence rates of the standard displacement error measured in the $L_2$ norm and $H^1$ seminorm. We observe that the variational embedding of data improves the accuracy of the solution while still retaining the variational consistency of the DG method. The $L_2$ norm of error is significantly reduced while the absolute error in the seminorm is reduced slightly. For the case where the data is embedded at location 2, $H^1$ seminorm of error is slightly higher than the case with no data. However, the convergence rate of the $H^1$ seminorm for both cases is 0.49 which is close to the rate predicted by the theory.

Table 2:

Description of the non-conforming mesh hierarchy of 4-node quadrilaterals.

Mesh	Quadrilateral (Q4)
	Elements	Nodes
Mesh 1	96	130
Mesh 2	384	450
Mesh 3	1536	1666
Mesh 4	6144	6402

Figure 8:

Convergence rates of standard displacement error for non-conforming Q4 meshes.

5.2. Discrepancy Modeling in Elastodynamics: Free Vibration Beam

The first test case is a cantilever beam that is fixed at the left edge. It is subjected to an initial displacement field and then allowed to undergo free vibration. The domain length is 16 mm, while its width is 4 mm. The Lamé parameters used are $\mu =800\text{GPa}$, $\lambda =1200\text{GPa}$ which correspond to a Young’s Modulus $E=2080\text{GPa}$ and a Poisson’s ratio $\nu =0.3$. The density is taken to be $\rho =2\times {10}^{-3}{\text{Ns}}^2/{\text{mm}}^4$. Plane strain condition is assumed enforced, standard displacement-based Galerkin method is employed, and 4-node elements are used with Gauss quadrature rule that is sufficiently high to exactly integrate all the terms. First, the response of the system with a mesh that is comprised of 16 × 16 4-node elements (Figure 9a) is obtained, which is then treated as the target response for the system. Then, a reduced resolution mesh that is comprised of 8 × 8 4-node elements is generated (Figure 9b). This mesh results in a stiffer response that is attributed to the inherent stiffness of lower-resolution meshes for bending-dominated cases. The test problems are run by the average acceleration method using a constant time step of 2.5 × 10⁻⁴ s for a total time $t=0.1\text{s}$.

Figure 9:

Schematic diagrams of (a) the target system, and (b) the modeled system with a data patch.

We start with an imperfect model with parameter discrepancy by considering that the information about problem physics is incomplete and that the correct value of the density is not known. Accordingly, to model parameter discrepancy, the density is reduced to half. Because of the lower mass in the system with discrepancy, its frequency increases, which shows as phase error with respect to the target response. We then embed the measured data in a small patch of the imperfect system as indicated in Figure 9b. This results in correcting the phase error, and the reconstructed displacement time history compares well with that of the target system as shown in Figure 10a. Moreover, time histories of velocity at point A and stress at point B that represent the derivative in time and derivative in space of the underlying displacement field, respectively, are also recovered as presented in Figure 10(b-c). Since the DDV method reconstructs the field and its derivatives, therefore, to compute the stress field – which is a derived field – we employ the Young’s modulus of the target system.

Figure 10:

Time history of (a) vertical displacement at point A, (b) vertical velocity at point A, and (c) axial stress at point B (for the target system, and the system with density discrepancy).

In addition to the density mismatch, we now enhance parameter discrepancy by increasing Young’s modulus to $2\times E$. This results in an even stiffer system that shows a larger phase error in the displacement field as compared to the case with only density discrepancy. The data is embedded in a small patch, as in the previous case, and the discrete system emanating from the DDV method results in a displacement time history that is without any phase error w.r.t. the target system (Figure 11a). Likewise, velocity time history at point A of the standard Galerkin system shows phase error as well as amplitude error, and the DDV method is able to correct both, thereby producing a response that closely matches the target system, as shown in Figure 11b. However, the amplitude error in stress time history at point B still persists as shown in Figure 11c.

Figure 11:

Time history of (a) vertical displacement at point A, (b) vertical velocity at point A, and (c) axial stress at point B (for the target system, and the system with parameter discrepancy in density and stiffness).

To add complexity to the imperfect model, we introduce inadequacy in the boundary conditions by imposing simply supported boundary conditions as opposed to the fixed boundary conditions of the target system. The bottom node ($x=0$, $y=0$) at the left edge (root section) is a pin support while the top node ($x=0$, $y=H$) along the left edge is a roller, thereby constraining the displacement in the axial direction. Furthermore, the parameter discrepancy in density and stiffness is also maintained. The displacement field of the resulting system has phase error while the velocity and stress fields have phase error as well as amplitude error as shown in Figure 12 (a-c). The DDV method successfully corrects the amplitude and phase errors in the primary fields, i.e., displacement and velocity fields, while correcting the phase error but not the amplitude error in the derived stress field.

Figure 12:

Time history of (a) vertical displacement at point A, (b) vertical velocity at point A, and (c) axial stress at point B (for the target system, and the system with boundary conditions inadequacy and parameter discrepancy in density and stiffness).

The deformed configuration of the system for three different time points is shown in Figure 13 (a-c). It can be seen that the imperfect modeled system produces a response which is very distinct from the target system, and the DDV method is able to compensate for the boundary condition inadequacy as well as parameter discrepancy in density and stiffness. This test case shows the robustness of the proposed DDV method that can reconstruct the response that matches the target system while compensating for the multiple parameter discrepancies and boundary condition inadequacy.

Figure 13:

Deformed configuration of the cantilever beam at (a) t = 0 s, (b) t = 0.019 s, and (c) t = 0.048 s

5.2.1. System energy analysis

Evaluation of strain and kinetic energy involves displacement and velocity fields respectively, and therefore these are integrated quantities that involve computation over the entire domain. The time histories of the system strain energy, kinetic energy and external work are shown in Figure 14 (a-c). In the evaluation of the strain energy for the target system the value of Young’s modulus is equal to $E$, while in the evaluation of the energy for the modeled system with discrepancy, Young’s modulus is equal to $2\times E$. Likewise, the kinetic energy for the imperfect modeled system has half the mass as compared to the target system. In Figure 14c, we employ the time history of the strain energy and kinetic energy that is based on the reconstructed displacement and velocity fields from the DDV method as a measure of the accuracy of the integrated quantities over the domain.

Figure 14:

System energy of the cantilever beam for (a) target system (b) modeled system with parameter discrepancy and boundary condition inadequacy, (c) imperfect system modeled with the DDV method.

The loss term in the DDV method gives rise to a force type term that goes onto the right-hand side and results in external work done on the system as shown in Figure 14c. Moreover, the determinant of the stiffness matrix of the DDV-corrected system is larger than that of the original system because of the additional positive definite term that gets embedded in the formulation. Consequently, the initial strain energy is higher and this is reflected in the upward shift of the energy evolution plot in Figure 14c.

After obtaining the DDV-corrected displacement field, we analyze the time history of the strain energy as a measure of the accuracy of an integrated quantity over the domain. The strain energy of the system is calculated element-wise as follows

$$\text{Strain energy}=\sum _{e=1}^{nel}\underset{{\text{Ω}}^e}{\int }\frac{1}{2}{\mathit{\sigma }}^e{\mathit{ϵ}}^{e}d{\text{Ω}}^e$$

The blue line in Figure 15 shows the variation in strain energy as a function of time for the target system. The green line represents the strain energy of the problem with parameter discrepancy and inadequate BC. Since the evaluation of strain energy requires specification of the modulus of elasticity of the material, for plotting purpose we use the value of the modulus of the target system in the evaluation of stress which is then embedded in the calculation of the strain energy. While the maximum strain energy is nearly the same for the two systems, one can clearly see that the two dynamical systems have very different energy signatures. With the DDV method, we recover the response of the target system, as shown by the red line with dots. This serves as a comprehensive comparison of the reconstructed field as the displacement field over the entire domain is used in this evaluation.

Figure 15:

Strain energy of the cantilever beam for (a) time history between 0 and 0.1s (b) zoomed view along the time axis (scaled between 0 and 0.035s).

Figure 16 shows the kinetic energy of the system as a function of time. We use the value of the density of the target system to eliminate parameter discrepancy in the evaluation of the kinetic energy thereby providing an integrated measure of the quality of the velocity field. One can see that the maximum value of the kinetic energy is approximately 3 Joules which is in the same range as the maximum ordinate of the strain energy shown in Figure 16. The general behavior of the change in kinetic energy as a function of time is well represented by the DDV method, however, small overshoots can be seen around the peaks. This test case shows that overall, one can recover another significant derived quantity of the target system using the DDV method.

Figure 16:

Kinetic energy of the cantilever beam for (a) time history between 0 and 0.1s (b) zoomed view along the time axis (scaled between 0 and 0.035s).

5.2.2. Morlet wavelet analysis of the systems

We have employed Morlet wavelet Transform (WT) [36, 37] to visualize the fundamental frequencies of the target system, imperfect modeled system, and the system modeled with the DDV method. The Morlet wavelet is a Gaussian-windowed complex sinusoid based on an algorithm described in [36, 37]. Figure 17 (a-c) presents the Morlet wavelet transform of the displacement field for the three cases. The WT contour plots (WT spectra) show the amplitude of the WT against frequency (vertical axis) and time (horizontal axis). The heavily shaded areas represent regions of high WT amplitude, while the lightly shaded areas represent regions of low amplitude. The wavelet transform shows the existence of a steady fundamental frequency that is higher for the imperfect model because parameter discrepancy results in a high-frequency response of the system, but is restored to the target case once the data is embedded.

Figure 17:

Morlet wavelet transform of the cantilever beam for (a) target system, (b) imperfect system with parameter discrepancy and boundary condition inadequacy, and (c) imperfect system modeled with the DDV method.

5.3. Cantilever Beam with Edge Shear

The third test case is a cantilever beam that is subjected to parabolically varying shear at its right and left edges as shown in Figure 18. The geometry and parameters are the same as those employed in the previous case. However, the current problem is driven by an applied force boundary condition. At the root section along the left edge, Dirichlet boundary condition is applied which allows warping of the cross-section. Further detail on this problem is given in [32]. The tractions at the left and right edges are given by the following expressions:

$${\mathbf{h}}_{(x=0)}=\left[\begin{array}{c}3000y\\ -\frac{375}{4}\left(4-y^2\right)\end{array}\right]{\mathbf{h}}_{(x=16)}=\left[\begin{array}{c}0\\ \frac{375}{4}\left(4-y^2\right)\end{array}\right]$$

Figure 18:

Schematic diagram of the cantilever beam subjected to edge shear.

The same three scenarios from the previous test case are considered here, i.e., a target model that is comprised of 16 × 16 4-node elements, an imperfect model comprised of an 8 × 8 mesh with parameter discrepancy in density, and an imperfect model comprised of an 8 × 8 mesh with density discrepancy but with data embedded in a patch as indicated in Fig. 18. The displacement time history plotted at the top right corner of the beam (point A) for the three scenarios is shown in Figure 19. The parameter discrepancy induces phase error which gets corrected using the DDV method, thereby producing results that closely match the target system. Some small deviation is seen in the reconstructed displacement field owing to multiple interacting modes that make the dynamics of the problem more involved.

Figure 19:

Time history of vertical displacement at point A for cantilever beam with edge shear.

The deformed configuration of the cantilever beam at three different time points is shown in Figure 20 (a-c), and is scaled up by a factor of 5 for easy visualization. The modeled system with density mismatch moves to a high-frequency range and shows an out-of-phase behavior. DDV method is able to correct the dynamics of this system and the modeled response shows similar spatial distribution as that of the target system.

Figure 20:

Deformed configuration of the cantilever beam with edge shear at (a) t = 0.005 s, (b) t = 0.0135 s, (c) t = 0.0365 s.

5.4. Vertical Structure under Periodic Ground Motion

This test case involves forced vibration of a vertical cantilever beam that is subjected to time-dependent body force which is proportional to the spatial distribution of the mass of the beam. It is a representative model problem of a structure that is undergoing seismic loading through horizontal ground motion. The geometry, boundary conditions, and problem parameters are the same as those employed in the free vibration problem from Section 5.2. However, instead of the initial displacement field, the beam is now subjected to a body force of the form:

$$\mathbf{f}(t)=\rho {\ddot{\mathbf{z}}}_g=\rho \left[\begin{array}{c}{\ddot{z}}_g\\ 0\end{array}\right]=\rho \left[\begin{array}{c}A\sin \left(w_{f}t\right)\\ 0\end{array}\right]=\rho \left[\begin{array}{c}A\sin \left(2\pi f_{f}t\right)\\ 0\end{array}\right]$$

where $A={10}^4$ and $f_f=70\text{Hz}$. This frequency of the forcing function was particularly chosen as it is close to the fundamental natural frequency of the target 16 × 16 beam case – about 82.5 Hz – which results in a beat phenomenon. Beat occurs when two close frequencies interact, creating a pulsating motion where the magnitude of the oscillation is in itself a sinusoidal function. The amplitude of the response is maximal when the two signals are in phase and thus overlap, while it is minimum when they are 180° out of phase, therefore canceling each other. Since the system is undamped, one is able to create a beat phenomenon using the frequency of the input excitation.

First, the target response of the system is obtained with a mesh that is comprised of 4-node elements (Figure 21a). Then, the reduced resolution mesh is generated (Figure 21b) which results in a stiffer response due to the inherent stiffness associated with the lower resolution meshes. In the imperfect model, we have parameter discrepancy in the form of density mismatch, i.e., density is reduced to half as compared to the target system. Since the body force is directly proportional to the density, the discrepancy in density results in an inaccurate forcing function. For the parameters chosen, the natural frequency of the imperfect system moves away from the forcing frequency to the point where the beat phenomenon disappears. As a result, the system shows a large amplitude and phase error with respect to the target response as shown in Figure 22 (a-c). We then embed the measured data in the form of a time-dependent displacement field from the target system in a small patch of the imperfect model via the DDV method as indicated in Fig. 21b. The data-driven method corrects the phase error and restores the beat phenomena, resulting in time histories of displacement, velocity, and stress fields in the modeled system that match the response of the target system.

Figure 21:

Schematic diagrams of (a) the target system, and (b) the modeled system with the data patch.

Figure 22:

Time history of (a) horizontal displacement at point A, (b) vertical velocity at point A, and (c) axial stress at point B (for vertical beam subjected to ground motion).

As a next case, the parameter discrepancy is enhanced by introducing a mismatch in both the density and the stiffness of the imperfect model i.e., $\overline{\rho }=\rho /2$ and $\overline{E}=2E$. In addition, we also account for the imperfect information on the boundary conditions, i.e., instead of the fixed boundary condition at the base, the vertical beam is considered simply supported at the base as shown in Figure 21b. This allows the base to undergo local warping and therefore behave less stiff as compared to the fixed base. The resulting modeled system suffers from both the phase error and the amplitude error. Once again the DDV method is able to recover the target response by compensating for parameter discrepancy and boundary conditions inadequacy as shown in Figure 23 (a-c).

Figure 23:

Time history of (a) horizontal displacement at point A, (b) vertical velocity at point A, and (c) axial stress at point B (for vertical beam subjected to ground motion).

The deformed shape of the target system, the imperfect system, and the DDV modeled system for three different time points is shown in Figure 24 (a-c). The plots show the out-of-phase response of the imperfect system due to the discrepancy in parameters and the boundary condition. This is corrected by the DDV method which produces results wherein the phase and amplitude match the target system.

Figure 24:

Deformed configuration of the vertical beam at (a) t = 0.005 s, (b) t = 0.0265 s, (c) t = 0.046 s.

The Morlet wavelet transform of the displacement field for the target system and the modeled system with and without DDV is presented in Figure 25 (a-c). One can notice the existence of the two close frequencies for the target system. For the imperfect system, the input signal at 70 Hz has a high wavelet amplitude, with a secondary frequency appearing near 160 Hz, thereby explaining the absence of the beat phenomenon. Embedding of data through DDV results in correcting the frequency response of the imperfect system.

Figure 25:

Morlet wavelet transform of the cantilever beam for (a) target system, (b) modeled system with parameter discrepancy and boundary condition inadequacy, and (c) system modeled with the DDV method.

The phase diagram is commonly used to study the characteristics of a dynamical system and it gives a qualitative picture of the system behavior over time. We have plotted the phase diagram with displacement and velocity as variables. Each point in the phase diagram represents a unique state of the system, defined by a specific combination of displacement and velocity wherein the path traced by these points over time is called a trajectory. The time-dependent force alters the energy of the system which is reflected in the phase diagram. The force increases the energy of the target system initially, resulting in the trajectories in the phase diagram that move outwards i.e., away from the origin. Subsequently, the force decreases the energy of the system, and the trajectories move inwards towards the origin as shown in Figure 26a.

Figure 26:

Phase diagram of the vertical beam: (a) target system, (b) imperfect system with parameter discrepancy and boundary condition inadequacy, (c) system modeled with the DDV method, and (d) all three cases presented together.

For the imperfect system, the trajectories are an order of magnitude lower as compared to the target system since it is much stiffer due to the multiple parameter and boundary condition discrepancies. This is also reflected in the subdued response of the displacement and velocity time histories of the modeled system in Figure 23 (a-b). The DDV method with the data patch corrects the trajectories of the modeled system as shown in Figure 26 (c-d), and the reconstructed trajectories closely match the target system.

A summary of the response of the system is shown in Figure 27. Our target system is a vertical beam that is comprised of a 16 × 16 mesh and is subjected to ground motion. Our modeled system is comprised of an 8 × 8 mesh and has a discrepancy in density and stiffness, along with boundary condition inadequacy. The response of this deficient system is very distinct from that of the target system. When the DDV method embeds the measured data over a small patch, the reconstructed dynamical response matches well with that of the target system. This highlights the ability of the proposed method to compensate for parameter discrepancy and boundary condition inadequacy in the response of the dynamical system.

Figure 27:

Summary of the DDV method applied to an imperfect model of a vertical beam undergoing ground motion excitation.

5.5. Model Discrepancy

Model discrepancy arises when the physics-based model is missing some dominant terms that are otherwise needed to accurately represent the intended physical phenomena. This test case is a numerical validation of the DDV method to account for missing physics in the model. The target problem is a cantilever beam that is undergoing a damped periodic oscillation. The damping matrix is Rayleigh damping, $\mathbf{C}=a\mathbf{M}+b\mathbf{K}$ [32] with parameters $a=0$ and $b=5\times {10}^{-5}$ which results in a damping ratio of 1.3% at the fundamental natural frequency. The beam, comprising of 16 × 16 4-node elements, is fixed at its left edge. It is subjected to an initial displacement field and then allowed to undergo free vibration as shown in Figure 9. Figure 28 shows the displacement time history where amplitude decay can clearly be seen. We consider this simulation as our target response which has the following discrete system of equations:

$$\mathbf{M}\ddot{\mathbf{u}}+\mathbf{C}\dot{\mathbf{u}}+\mathbf{K}\mathbf{u}=\mathbf{F}$$ (49)

Figure 28:

Time history of the vertical displacement at point A showing damped response of the target system.

We now work with a model where the term corresponding to structural damping is not present. The matrix form of equations for this undamped system is as follows:

$$\mathbf{M}\ddot{\mathbf{u}}+\mathbf{K}\mathbf{u}=\mathbf{F}$$ (50)

The missing damping term in the above equation is a representative case of model discrepancy. Our objective is to see if the DDV method can compensate for missing physics in the computational framework.

Accordingly, data from the damped target system is embedded in the undamped model via the DDV method. The response of the resulting system closely matches the damped response as shown in Figure 29 (a-c). We have also plotted the velocity time history and the stress time history, and once again we see that the DDV method successfully embeds high-fidelity data from a damped system to correct the dynamical response of the deficient (undamped) physical model.

Figure 29:

Time history of (a) vertical displacement at point A, (b) vertical velocity at point A, and (c) axial stress at point B (for cantilever beam with Rayleigh damping).

We now increase the level of deficiency in the system and add parameter discrepancy as well as boundary condition inadequacy in addition to the model discrepancy. Accordingly, the density of the imperfect system is halved while its stiffness is doubled as compared to that of the target system. Moreover, the boundary conditions at the left edge allow local warping of the cross-section as opposed to being fixed. In addition, there is no physical damping in the imperfect system. The DDV method with a small data patch is able to successfully correct phase and amplitude errors in the imperfect system and produce a response that matches well with that of the target system that has physical damping as shown in Figure 30 (a-c).

Figure 30:

Time history of (a) vertical displacement at point A, (b) vertical velocity at point A, and (c) axial stress at point B (for the cantilever beam with model discrepancy, parameter discrepancy, and boundary condition inadequacy).

The strain energy is an integrated quantity that involves the displacement field over the entire domain. In order to analyze the quality of this spatially distributed displacement field, we plot the time history of the strain energy in Figure 31. The blue line shows the variation in strain energy as a function of time for the target system. Because of damping, the peak value for each cycle uniformly decreases with time. The green line represents the strain energy of the imperfect system with the model discrepancy, parameter discrepancy, and inadequate BC. We use the value of the modulus of the target system in the evaluation of stress which is then embedded in the calculation of the strain energy. As this modeled system lacks damping, the maximum value at each cycle remains the same. Additionally, the period of the cycles is different when compared with the target system. With the DDV method, we recover the response of the target system, as demonstrated by the red line. This serves as a more comprehensive comparison as it involves the constructed displacement field over the entire domain in the evaluation of strain energy.

Figure 31:

Strain energy of the cantilever beam for (a) time history between 0 and 0.1s (b) zoomed view along the time axis (scaled between 0 and 0.035s).

Figure 32 shows the kinetic energy of the system as a function of time and it involves the velocity field over the domain. We use the value of density of the target system in the evaluation of kinetic energy so that the quality of the DDV-based velocity field can be compared with the target system. Once again, we see a similar behavior of the kinetic energy of the DDV formulation as was observed in Figure 16.

Figure 32:

Kinetic energy of the cantilever beam for (a) time history between 0 and 0.1s (b) zoomed view along the time axis (scaled between 0 and 0.035s).

6. Conclusion

We have presented a data-driven framework that embeds measured data in forward calculations to model discrepancies between observations and simplified or deficient physical models. A variationally consistent derivation of the method is presented wherein a loss function emerges that is based on the difference between the computed solution and the target value. This loss function penalizes the distance between the computed solution and the target solution, thereby minimizing the error in the solution space.

The method is implemented in the context of finite elements, employing the standard Galerkin form in space and the average acceleration method in time. The variational construct of the DDV method exploits features of the discontinuous Galerkin method for weak enforcement of continuity of the solution in the spatial domain, in addition to enforcing the match between the modeled response and the target value via loss function, which results in a reconstructed field that compensates for parameter and model discrepancies.

To analyze the method and numerically validate its variational consistency, we employ a problem that has an exact solution but also has a weak singularity to represent engineering problems of practical interest.

We then employ simple problems from elastodynamics as model cases to investigate the discrepancy modeling feature of the method. The first test case is a cantilever beam under free vibration, that is driven by a specified initial displacement field. We create the imperfect model that suffers from parametric discrepancy and boundary condition inadequacy, and then employ the DDV method to reconstruct the time-dependent displacement and velocity fields. Comparison of displacement and velocity time histories at select, but otherwise arbitrary spatial locations, as well as comparisons of strain and kinetic energies which represent integrated quantities based on the reconstructed displacement and velocity fields show that the DDV method can effectively compensate for multiple discrepancies in the reconstructed dynamical system. The wavelet transform method is employed to investigate the frequency spectrum of the reconstructed system with parameter and BC discrepancies. It is shown that the frequency spectrum for the imperfect model, when reconstructed via the DDV method, is able to recover the frequency spectrum of the target dynamic system.

The method is applied to an engineering problem that arises in the field of earthquake engineering and it investigates the effect of time-dependent body force on the dynamics of the system. The phase diagram between displacement and velocity for the target system and for the imperfect system with discrepancies shows a large difference for the two cases. The proposed DDV method successfully reconstructs the dynamical system that compares well with the target system.

The method is applied to a representative case of model discrepancy with missing physics. We reconstruct the dynamic response of a damped physical system while working with a reduced surrogate system that does not have the terms that can model physics-based damping. This test case verifies that the DDV method can successfully account for missing physics and therefore adequately address model discrepancy and inadequacy.

Technical Highlights.

1. A data-driven discrepancy modeling method that variationally embeds measured data in the modeling and analysis framework.

2. First-principle based model is augmented with variationally derived loss function that results in forward simulations of dynamical systems wherein high-fidelity sparse data compensates for the model discrepancy.

3. Numerical test cases produce theoretically predicted rates for the convergence of the solution, thereby validating variational consistency of the method.

4. Response of a damped physical system is reconstructed while working with a physics-based model with missing physical damping along with parameter discrepancy and boundary condition inadequacy.

Appendix A. Implementation of the DDV Method

Dropping flux data terms and simplifying equ. (35) for numerical implementation yields the following DDV formulation,

$$\begin{gathered} \mathcal{A}{\left({\mathbf{W}}_1,{\mathbf{U}}_1\right)}_{{\text{Ω}}_1}+\mathcal{A}{\left({\mathbf{W}}_2,{\mathbf{U}}_2\right)}_{{\text{Ω}}_2}+{\langle {\mathbf{w}}_1,{\left({\tau }_1^D\right)}^{-1}\left({\mathbf{u}}_1-{\mathbf{u}}^{(data)}\right)\rangle }_{{\text{Γ}}_I} \\ +{\mathbf{w}}_2,{\left({\tau }_2^D\right)}^{-1}{\left({\mathbf{u}}_2-{\mathbf{u}}^{(data)}\right)}_{{\text{Γ}}_I}=\mathcal{L}{\left({\mathbf{W}}_1\right)}_{{\text{Ω}}_1}+\mathcal{L}{\left({\mathbf{W}}_2\right)}_{{\text{Ω}}_2} \end{gathered}$$ (A.1)

Substituting the domain interior integrals, the formulation can be written as:

$$\begin{gathered} \sum _{\alpha =1,2}\left[{\left({\mathbf{w}}_{\alpha },{\rho }_{\alpha }{\ddot{\mathbf{u}}}_{\alpha }\right)}_{{\text{Ω}}_{\alpha }}+{\left(ϵ\left({\mathbf{w}}_{\alpha }\right),\mathbf{\sigma }\left({\mathbf{u}}_{\alpha }\right)\right)}_{{\text{Ω}}_{\alpha }}\right]+\sum _{\alpha =1,2}\left[{\langle {\mathbf{w}}_{\alpha },{\left({\tau }_{\alpha }^D\right)}^{-1}\left({\mathbf{u}}_{\alpha }-{\mathbf{u}}^{(data)}\right)\rangle }_{{\text{Γ}}_I}\right] \\ =\sum _{\alpha =1,2}\left[{\left({\mathbf{w}}_{\alpha },{\mathbf{b}}_{\alpha }\right)}_{{\text{Ω}}_{\alpha }}+{\left({\mathbf{w}}_{\alpha },{\mathbf{h}}_{\alpha }\right)}_{{\text{Γ}}_{\alpha }}\right] \end{gathered}$$ (A.2)

From equ. (A.2), we now write the expansions in terms of element shape functions to help with the numerical implementation of the method.

Since the explicit expressions naturally segregate into two subdomains having the same equations, with no cross terms, we focus on one side only.

The interface component of the residual term is expressed as follows:

$$\begin{gathered} {\mathbf{R}}_{\alpha }^{Interface}=-{\mathcal{B}}^{Interface}\left({\mathbf{w}}_{\alpha },{\mathbf{u}}_{\alpha }\right) \\ =-{\langle {\mathbf{w}}_{\alpha },{\mathit{\tau }}_{\alpha }^D\left({\mathbf{u}}_{\alpha }-{\mathbf{u}}^{(data)}\right)\rangle }_{{\text{Γ}}_I} \\ =-\underset{{\text{Γ}}_I}{\int }{\mathbf{w}}_{\alpha }\cdot {\mathit{\tau }}_{\alpha }^D\left({\mathbf{u}}_{\alpha }-{\mathbf{u}}^{(data)}\right)da \end{gathered}$$

We employ the standard shape functions,

$${\mathbf{u}}_{\alpha }^e=\sum _{a=1}^{nel}{\left(N^a\right)}^e{\left({\mathbf{u}}_{\alpha }^a\right)}^e={\left(N^a\right)}^e{\left({\mathbf{u}}_{\alpha }^a\right)}^e\text{(using index notation)}$$

For ease of notation, we are dropping the explicit appearance of $\alpha$ and $e$ while it is implied that the following quantities are evaluated at the element level, and for domains 1 and 2. As such, the element-level quantities can be written as

$$\begin{gathered} {\mathbf{R}}^{Interface}=-\underset{{\text{Γ}}_I}{\int }{N}^a{\mathbf{w}}^a\cdot {\mathit{\tau }}^D\left(\mathbf{u}-{\mathbf{u}}^{(data)}\right)da \\ =-\underset{{\text{Γ}}_I}{\int }{N}^a{w}_i^a{\tau }_{ij}^D\left(u_j-u_j^{(data)}\right)da \end{gathered}$$

As such, at nodal point $a$, and for degree of freedom $i$, the residual is expressed as:

$${\left(R^{Interface}\right)}_i^a=-\underset{{\text{Γ}}_{\text{I}}}{\int }{N}^a{\tau }_{ij}^D\left(u_j-u_j^{(data)}\right)da$$

To get the tangent, we employ a variational derivative of ${\mathcal{B}}^{Interface}\left({\mathbf{w}}_{\alpha },{\mathbf{u}}_{\alpha }\right)$ with respect to ${\mathbf{u}}_{\alpha }$

$$\begin{gathered} {\mathbf{K}}_{\alpha }^{Interface}=D_{{\mathbf{u}}_{\alpha }}\left[{\mathcal{B}}^{Interface}\left({\mathbf{w}}_{\alpha },{\mathbf{u}}_{\alpha }\right)\right]\cdot \text{Δ}{\mathbf{u}}_{\alpha }={\langle {\mathbf{w}}_{\alpha },{\mathit{\tau }}_{\alpha }^D\text{Δ}{\mathbf{u}}_{\alpha }\rangle }_{{\text{Γ}}_I} \\ =\underset{{\text{Γ}}_{\text{I}}}{\int }{\mathbf{w}}_{\alpha }\cdot {\mathit{\tau }}_{\alpha }^D\text{Δ}{\mathbf{u}}_{\alpha }da \end{gathered}$$

Again, employing standard shape functions, and dropping the explicit appearance of $\alpha$ and $e$ leads to the following expression,

$$\begin{gathered} {\mathbf{K}}^{Interface}=\underset{{\text{Γ}}_{\text{I}}}{\int }{N}^a{\mathbf{w}}^a\cdot {\mathit{\tau }}^D{N}^b\text{Δ}{\mathbf{u}}^{b}da \\ =\underset{{\text{Γ}}_{\text{I}}}{\int }{N}^a{w}_i^a{\tau }_{ij}^D{N}^b\text{Δ}{u}_j^{b}da \end{gathered}$$

As such, for nodal points $a$ and $b$, and for the ($i$, $j$) component, the tangent is expressed as:

$${\left(K^{Interface}\right)}_{ij}^{ab}=\underset{{\text{Γ}}_{\text{I}}}{\int }{N}^a{\tau }_{ij}^D{N}^{b}da$$