Model-based simulations of pulsed laser ablation using an embedded finite element method

Abstract

A model of thermal ablation with application to multi-pulsed laser lithotripsy is presented. The approach is based on a one-sided Stefan–Signorini model for thermal ablation, and relies on a level-set function to represent the moving interface between the solid phase and a fictitious gas phase (representing the ablated material). The model is discretized with an embedded finite element method, wherein the interface geometry can be arbitrarily located relative to the background mesh. Nitsche’s method is adopted to impose the Signorini condition on the moving interface. A bound constraint is also imposed to deal with thermal shocks that can arise during representative simulations of pulsed ablation with high-power lasers. We report simulation results based on experiments for pulsed laser ablation of wet BegoStone samples treated in air, where Begostone has been used as a phantom material for kidney stone. The model is calibrated against experimental measurements by adjusting the percentage of incoming laser energy absorbed at the surface of the stone sample. Simulation results are then validated against experimental observations for the crater area, volume, and geometry as a function of laser pulse energy and duration. Our studies illustrate how the spreading of the laser beam from the laser fiber tip with concomitantly reduced incident laser irradiance on the damaged crater surface explains trends in both the experimental observations and the model-based simulation results.

1. Introduction

Model-based simulations of laser ablation are of great interest and utility for engineering and medical applications. Multi-pulsed laser ablation, describing the removal of matter from material surfaces with multiple laser pulses, has motivated experimental [6,20,22,27,29] and theoretical research [1,19,21,24,27]. Applications of laser ablation range from materials processing technology to medical treatments, with some recent research focusing on laser lithotripsy [9,10,25,28]. Regardless of the particular application, the physical processes involved in laser ablation are quite complex, ranging from light absorption to heat conduction to phase transformation and material removal. As a result, high-fidelity numerical simulations of laser ablation can be challenging to obtain.

Previous work on laser ablation simulations and lithotripsy tend to focus on a limited number of aspects of the complex multi-pulse laser ablation process. Wood and Geist [26] proposed a computational model for melting and solidification in pulsed-laser-irradiated materials. Anisimov and Luk’yanchuk [1] discussed the mechanical model and the dynamic model of the laser ablation theory. Fell et al. [12] developed a numerical application to simulate heating, phase changes, and material removal during the pulsed laser ablation. Korfiatis et al. [19] also worked on the numerical modeling of ultrashort-pulse laser ablation of silicon. Li et al. [22] performed a computational and experimental study of nanosecond laser ablation of crystalline silicon. Recently, Wu et al. [27] discussed the reflection and absorption of a pulsed laser on a material surface.

In this work, an energetic model is developed based on the one-sided Stefan–Signorini model for ablation and an embedded finite element discretization, as described in Claus et al. [8]. In the one-sided Stefan problem, the interface essentially represents the evolving boundary separating intact from removed (or fictitious) material. The interface is advected into the physical domain as a function of the difference between the incoming laser beam energy and the thermal resistance of the material, as well as the latent heat. Although the use of a discretized Stefan–Signorini model for ablation is relatively new, it bears emphasis that the underlying model is mostly energetic in nature. It neglects a range of physical phenomena that are known to occur in the ablation process, including laser scattering, the formation of a melt pool on the surface, and chemo-mechanical effects. Nevertheless, due to its very simplicity the calibration of this model against experiments of laser ablation sheds insight into an energetic accounting of ablation.

In terms of discretization, the model presents three challenges that are non-standard to varying degrees. These include: 1) the need to represent an evolving interface geometry; 2) the need to calculate fields on just one side of the interface; and 3) the need to impose appropriate constraints on the interface. Claus et al.[8] addressed these challenges by employing a level-set method to capture the interface geometry, combined with a Cut-FEM scheme [16] to allow the interface geometry to be arbitrary relative to the mesh. Interfacial constraints were imposed weakly using a variation of Nitsche’s method. The capabilities of the model were demonstrated with various two- and three-dimensional simulations of ablation in Claus et al.[8].

In this work, we further refine the method to simulate recent experiments of pulsed laser ablation of BegoStone. BegoStone is a commercially available high strength plaster of Paris that has been used as a phantom material for lithotripsy research [23]. As laser lithotripsy continues to increase as a treatment protocol, so too does fundamental research to support it. But the actual material properties of BegoStone combined with the particular laser that was employed in the experiments give rise to highly localized temperature gradients where the beam meets the surface. At early times a type of “thermal shock” can arise that can trigger spurious oscillations in the simulated temperature fields, even when fairly refined meshes are employed [11]. To address this challenge, we have modified the method of Claus et al.[8] to work for axisymmetric problems, enabling the use of highly refined meshes. We have also incorporated a bound constraint on the discrete temperature fields so that they remain within physically-reasonable limits, even when a very large heat flux is simulated.

This paper is organized as follows. In Section 2, we first present the axisymmetric model for pulsed laser ablation, the governing equations with the Signorini conditions, and the laser beam configuration. In addition, the level set function describing the interface is introduced. In Section 3, the discretization of the embedded finite element method is presented with an accompanying nonlinear solution strategy. In Section 4, experimental results for pulsed laser ablation of treated BegoStone samples are shown and discussed. In Section 5, the proposed model is calibrated to the experimental results by adjusting the assumption for the incoming laser energy absorption. The simulation results based on experiments are reported here in detail. Finally, model-based simulation results are validated against experimental observations for the crater surface profile area, volume, and geometry as a function of laser pulse energy and duration. The results are discussed and we also provide, followed by a summary and some concluding remarks at the end of the manuscript.

2. Problem formulation

In this section, an axisymmetric model of laser ablation is developed in detail. Starting from a fully three-dimensional model of laser ablation, the configuration for a two-dimensional axisymmetric problem is developed. Then the interfacial conditions are described, and the profile for a representative incoming laser beam is provided.

2.1. The boundary value problem

A schematic for an axisymmetric model of a thermal ablation problem is given in Fig. 1. The physical model is essentially a Stefan problem with a moving boundary or ablation surface where Signorini boundary conditions are imposed [13]. Specifically, the material domain $\Omega (t)$ in ${ℝ}^d$ (d = 2, 3) is assumed to be heated from above by a laser beam that results in material being ablated from the surface. As such, the top surface Γ(t), with unit outward normal n_Γ, represents the interface between the intact material and anything that is removed via thermal ablation. The remaining portions of the boundary are separated into Neumann $\partial {\Omega }_N$ and Dirichlet $\partial {\Omega }_D$ parts with unit outward normal n_Ω.

Fig. 1.

Schematic of (a) a two-dimensional axisymmetric laser ablation problem, (b) the corresponding three-dimensional geometry within a cylindrical domain.

The temperature field T in the domain is assumed to be governed by the heat equation

$$\rho c_p\frac{\partial T}{\partial t}-\kappa \Delta T=f\text{in}\Omega (t),\text{for}t\in \left[t_0,t_f\right]$$ (1)

where c_p is the specific heat capacity, ρ is the mass density, κ is the thermal conductivity, and f denotes a source term. The Dirichlet and Neumann boundary conditions are given by

$$\begin{array}{cc}T& =T_D\text{on}\partial {\Omega }_D(t),\\ \kappa \nabla T\cdot {\mathit{n}}_{\Omega }& =q_N\text{on}\partial {\Omega }_N(t),\end{array}$$ (2)

and the initial condition is

$$T\left(\mathit{x},t_0\right)=T_0\text{in}\Omega \left(t_0\right).$$ (3)

The interface Γ(t) evolves with velocity v (pointing inward as shown in Fig. 1(a)) in proportion to the difference between the incoming energy (heat flux) I on the surface and the thermal resistance of the material. In particular, the following interfacial energy balance is assumed to hold:

$$I(\mathit{x},t)\cdot {\mathit{n}}_{\Gamma }-\kappa \nabla T\cdot {\mathit{n}}_{\Gamma }=-\rho L_t\left(\mathit{v}(\mathit{x},t)\cdot {\mathit{n}}_{\Gamma }\right)\text{on}\Gamma (t),$$ (4)

where L_t denotes the latent heat for melting.

In addition to the above, the following Signorini conditions hold on the interface:

$$\begin{array}{ccc}\kappa \nabla T\cdot {\mathit{n}}_{\Gamma }-I\cdot {\mathit{n}}_{\Gamma }\le 0& \text{on}& \Gamma (t),\\ T-T_m\le 0& \text{on}& \Gamma (t),\\ \left(\kappa \nabla T\cdot {\mathit{n}}_{\Gamma }-I\cdot {\mathit{n}}_{\Gamma }\right)\perp \left(T-T_m\right)& \text{on}& \Gamma (t).\end{array}$$ (5)

In essence, these conditions require that material is only removed if the temperature reaches the melting temperature T_m, and that material is only removed and not added to the surface. Further, they stipulate that the temperature not exceed the melting temperature, e.g. if ablation is performed, then it is performed at constant interface temperature T_m.

2.2. Weak form

The weak form of the Stefan–Signorini problem is: $\forall t\in \left[t_0,t_f\right]$, find $T\in 𝒰$ such that

$$\underset{\Omega (t)}{\int }w\rho c_p\frac{\partial T}{\partial t}\text{dV}+\underset{\Omega (t)}{\int }\nabla w\cdot \kappa \nabla T\text{dV}=\underset{\Omega (t)}{\int }wf\text{dV}+\underset{\Gamma (t)}{\int }w\kappa \nabla T\cdot {\mathit{n}}_{\Gamma }\text{dA}+\underset{\partial {\Omega }_N(t)}{\int }w{q}_N\text{dA},\forall w\in 𝒱$$ (6)

where the trial and the test spaces are

$$\begin{array}{l}𝒰=\left\{T\in {\mathscr{H}}^1(\Omega (t))\mid T=T_D\text{on}\partial {\Omega }_D(t)\right\},\\ 𝒱=\left\{w\in {\mathscr{H}}^1(\Omega (t))\mid w=0\text{on}\partial {\Omega }_D(t)\right\}.\end{array}$$ (7)

To implement the Signorini conditions (5), an auxiliary interface variable defined over Γ(t) is introduced by

$$\sigma ≔\kappa \nabla T\cdot {\mathit{n}}_{\Gamma }-I\cdot {\mathit{n}}_{\Gamma },$$ (8)

so that the weak form becomes

$$\underset{\Omega (t)}{\int }w\rho c_p\frac{\partial T}{\partial t}\text{dV}+\underset{\Omega (t)}{\int }\nabla w\cdot \kappa \nabla T\text{dV}=\underset{\Omega (t)}{\int }wf\text{dV}+\underset{\partial {\Omega }_N(t)}{\int }w{q}_N\text{dA}+\underset{\Gamma (t)}{\int }w\sigma \cdot {\mathit{n}}_{\Gamma }\text{dA}+\underset{\Gamma (t)}{\int }wI\cdot {\mathit{n}}_{\Gamma }\text{dA},$$ (9)

and the Signorini condition (5) becomes

$$\begin{array}{ccc}\sigma \le 0& \text{on}& \Gamma (t),\\ T-T_m\le 0& \text{on}& \Gamma (t),\\ \sigma \perp \left(T-T_m\right)& \text{on}& \Gamma (t).\end{array}$$ (10)

2.3. Nitsche reformulation

We follow [4,7] and reformulate (10) as

$${\sigma }_{\gamma }=-\frac{1}{\gamma }{\langle \left(T-T_m\right)-\gamma \sigma \rangle }_{+},$$ (11)

where γ > 0 is a penalty parameter and ${\langle a\rangle }_{+}≔(|a|+a)/2$. Rewriting σ entirely in terms of temperature T now yields

$${\sigma }_{\gamma }(T)=-\frac{1}{\gamma }{\langle \left(T-T_m\right)-\gamma \left(\kappa \nabla T\cdot {\mathit{n}}_{\Gamma }-I\cdot {\mathit{n}}_{\Gamma }\right)\rangle }_{+}.$$ (12)

The penalty form to weakly enforce σ_γ (T) ≤ 0 is formulated by

$$s(w,T)≔\underset{\Gamma (t)}{\int }\left({\theta }_{1}w-{\theta }_2\gamma \kappa \nabla w\cdot {\mathit{n}}_{\Gamma }\right)(\left(\kappa \nabla T\cdot {\mathit{n}}_{\Gamma }-I\cdot {\mathit{n}}_{\Gamma }\right)-{\sigma }_{\gamma }(T))\text{d}\Gamma ,$$ (13)

where the parameters θ₁ ∈ {0, 1} and θ₂ ∈ {−1, 0, 1} lead to various Nitsche methods. In this work, we choose the semi penalty-free Nitsche method [4], where θ₁ = 0 and θ₂ = −1.

In the following, in order to simplify the presentation, we use (·)_Ω to denote the ${ℒ}_2$ inner product on Ω. The Stefan–Signorini–Nitsche formulation can then be written as

$$a(w,T)-{\left(w,\kappa \nabla T\cdot {\mathit{n}}_{\Gamma }-I\cdot {\mathit{n}}_{\Gamma }\right)}_{\Gamma (t)}+s(w,T)=l(w),$$ (14)

where a(w, T) and l(w) denotes the bilinear form and linear form, respectively

$$a(w,T)≔\rho c_p{(w,\frac{\partial T}{\partial t})}_{\Omega (t)}+{(\nabla w,\kappa \nabla T)}_{\Omega (t)},$$ (15)

$$l(w)≔{(w,f)}_{\Omega (t)}+{\left(w,q_N\right)}_{\Omega (t)}+{\left(w,I\cdot {\mathit{n}}_{\Gamma }\right)}_{\Gamma (t)}.$$ (16)

In order to simplify the formulation, P_γ(T) and $P_{\theta \gamma }^{\delta }(w)$ are defined as

$$P_{\gamma }(T)≔\left(T-T_m\right)-\gamma \left(\kappa \nabla T\cdot {\mathit{n}}_{\Gamma }-I\cdot {\mathit{n}}_{\Gamma }\right),$$ (17)

$$P_{\theta \gamma }^{\delta }(w)≔{\theta }_{1}w-\gamma {\theta }_2\kappa \nabla w\cdot {\mathit{n}}_{\Gamma },$$ (18)

so that the penalty form reads

$$s(w,T)≔\underset{\Gamma (t)}{\int }{P}_{\theta \gamma }^{\delta }(w)((\kappa \nabla T\cdot {\mathit{n}}_{\Gamma }-I\cdot {\mathit{n}}_{\Gamma })+\frac{1}{\gamma }\langle P_{\gamma }{(T)\rangle }_{+})\text{d}\Gamma .$$ (19)

Finally, the Nitsche formulation reads

$$a(w,T)+{(P_{\theta \gamma }^{\delta }(w)-w,\kappa \nabla T\cdot {\mathit{n}}_{\Gamma })}_{\Gamma (t)}+𝒩(w,T)=l(w)+{(P_{\theta \gamma }^{\delta }(w)-w,I\cdot {\mathit{n}}_{\Gamma })}_{\Gamma (t)},$$ (20)

where

$$𝒩(w,T)=\frac{1}{\gamma }{(P_{\theta \gamma }^{\delta }(w),\langle P_{\gamma }{(T)\rangle }_{+})}_{\Gamma }(t),$$ (21)

with γ a scaling parameter that enforces the Neumann condition at the interface when T < T_m.

2.4. Level set function and the ablated volume

The moving interface Γ(t) (the red line/ surface in Fig. 1) is represented by a level-set function $\varphi :\Omega (t)\to ℝ$, where ϕ(x, t) = 0 describes the location of the interface. The material domain Ω and the ablated domain are defined by ϕ(x, t) < 0 and ϕ(x, t) > 0, respectively, as shown in Fig. 1.

In ${ℝ}^3$, the volume V_a of the ablated crater can be calculated simply using

$$V_a=\int \underset{\Omega }{\int }\int \mathscr{H}(\varphi (\mathit{x}))\text{d}\mathit{x},$$ (22)

where $\mathscr{H}(\cdot )$ is the Heaviside function. In a two-dimensional axisymmetric model (x → (r, z)), (22) simplifies to

$$V_a=\int \underset{\Omega }{\int }\int \mathscr{H}(\varphi (r,z))r\text{d}r\text{d}z\text{d}\theta .$$ (23)

In a discrete setting, these integrals are approximated with numerical quadrature. Details are provided in Section 3.

2.5. Laser beam profile

We assume a laser beam that is directed downward towards the domain Ω from above, in the direction e_ray = {0, −1}^T. The laser beam for the axisymmetric model is adapted from one with a spatial profile that has a Gaussian distribution (Fig. 2(a)), viz:

$$I(r,t)=-f(r,t){\mathit{e}}_{ray},$$ (24)

$$f(r,t)=f_r(r,t)f_t(t)≔A_{amp}\frac{1}{\sqrt{{\left(2\pi {\sigma }^2\right)}^{d-1}}}{e}^{\frac{-\parallel x{\parallel }^2}{2{\sigma }^2}}{f}_t(t),$$ (25)

Fig. 2.

Laser pulse setting: (a) spatial profile, (b) temporal periodic switch function.

where σ is the radius of the beam, A_amp is the amplitude of the beam, and f_t (t) characterizes the temporal behavior of the laser pulse. For example, f_t (t) ≡ 1 would correspond to a laser beam that remains constant in time.

In this work, we will be interested in modeling laser pulses with a periodic evolution in time, cycling between on and off. This can be effected by assuming the following form for f_t (t) as Fig. 2(b) shows:

$$f_t(t)=\{\begin{array}{l}1\text{if}t-\lfloor \frac{t}{P_0}\rfloor P_0\le t_{on},\hfill \\ 0\text{else},\hfill \end{array}$$ (26)

where P₀ denotes the magnitude of the time period and $t_{\mathit{\text{on}}}\le P_0$ the on time, and ⌊·⌋ denotes the floor operator.

To match the laser energy input between a two-dimensional axisymmetric model and a general three-dimensional model, a factor of $\sqrt{2\pi }\sigma$ (from (25)) is needed:

$$\underset{0}{\overset{2\pi }{\int }}\underset{\Gamma }{\int }{I}^{2D}r\text{d}r\text{d}\theta =\underset{\Gamma }{\int }{I}^{3D}\text{dA}\to A_{amp}^{2D}=A_{amp}^{3D}/\sigma \sqrt{2\pi }.$$ (27)

3. Discretization

In this section, the discretization in space and time is presented. We employ an embedded finite element method in space and an implicit integrator in time. Details are provided in the subsections that follow.

3.1. Galerkin approximation

Firstly, we introduce important domains in this work. Let Ω(t₀ ) be the material domain at t = t₀ as Fig. 1(a) shows, where Ω(t) is defined by ϕ(x, t) < 0. We write Γ_h and Ω_h to define the integration domain. These are linear approximations of the interface Γ and the material domain Ω as shown in Fig. 4. A two-grid solution proposed by Gross et al. [14], Gross and Reusken [15] is adopted to define the discrete domain Ω_h and Γ_h. The temperature is approximated with linear finite elements (31), while the level set function is approximated with quadratic finite elements (41). The mesh in the interface region is refined to compute a piece-wise linear approximation of the boundary of the quadratic level set in each element. Details about the discretization of the level set function and mesh refinement are introduced in Sections 3.2 and 3.3.

Fig. 4.

Illustration of linear approximation Γ_h to interface Γ and the discrete integration domain Ω_h. The thick black lines represent the original mesh, the thin gray lines represent the refined background mesh, and the shaded part denotes the integration domain after mesh refinement and cutting.

In the embedded finite element method, a fixed background domain Ω_b is defined such that

$${\Omega }_b=\underset{K\in {\widetilde{𝒯}}_h}{\cup }K,$$ (28)

where ${\widetilde{𝒯}}_h$ is the background mesh which remains fixed in time. The active mesh is defined as

$${𝒯}_h(t)=\left\{K\in {\widetilde{𝒯}}_h\mid K\cap \Omega (t)\ne \varnothing \right\},$$ (29)

representing elements that are contained either completely or partially in Ω(t) (the gray and green shaded elements in Fig. 3). In Fig. 3, elements with green edges correspond to the interface layer.

Fig. 3.

Schematics of the fixed background domain Ω_b, the time-evolving active domain Ω*(t), the interface Γ(t), and the ghost-penalty faces ${ℱ}_{\Gamma (t)}$.

The active domain is defined as the union of all elements in active mesh ${𝒯}_h(t)$ as

$${\Omega }^{\ast }(t)=\underset{K\in {𝒯}_h(t)}{\cup }K.$$ (30)

The finite element space on the active domain is defined as

$${𝒱}_h(t)=\left\{T_h\in {𝒞}^0\left({\Omega }^{\ast }(t)\right)\right\},$$ (31)

consisting of piecewise-polynomial functions constructed over each of the elements. In particular, we use piecewise-linear functions.

Recalling (20), the Galerkin form of the problem can then be written as: $\forall t\in \left[t_0,t_f\right]$, find $T_h\in {𝒱}_h(t)$ such that

$$A\left(w_h,T_h\right)+𝒩\left(w_h,T_h\right)=L\left(w_h\right)\forall w_h\in {𝒱}_h(t),$$ (32)

where

$$\begin{array}{cc}A(w_h,T_h)& =a(w_h,T_h)+a_{bc}(w_h,T_h)+s_T(w_h,T_h)+{(P_{\theta \gamma }^{\delta }(w_h)-w_h,\kappa \nabla T_h\cdot {\mathit{n}}_{\Gamma })}_{{\Gamma }_h(t)},\\ L(w_h)& =l(w_h)+{(P_{\theta \gamma }^{\delta }(w_h)-w_h,I\cdot {\mathit{n}}_{\Gamma })}_{{\Gamma }_h(t)},\hfill \\ 𝒩(w_h,T_h)& =\frac{1}{\gamma }{(P_{\theta \gamma }^{\delta }(w_h),P_{\gamma }{(T_h)}_{+})}_{{\Gamma }_h(t)},\hfill \end{array}$$ (33)

and with the initial condition

$$T_h(\mathit{x},t_0)=T_0{\text{in}\Omega }_h(t_0).$$ (34)

In the above, the bilinear term, linear term, and boundary terms on Γ(t) are given by

$$\begin{array}{cc}a(w_h,T_h)& =\rho c_p{(w_h,\frac{\partial T_h}{\partial t})}_{{\Omega }_h(t)}+{(\nabla w_h,\kappa \nabla T_h)}_{{\Omega }_h(t)},\hfill \\ l(w_h)& ={(w_h,f)}_{{\Omega }_h(t)}+{(w_h,q_N)}_{\partial {\Omega }_N(t)}+{(w_h,I\cdot {\mathit{n}}_{\Gamma })}_{{\Gamma }_h(t)},\hfill \\ a_{bc}(w_h,T_h)& =-{(w_h,\kappa \nabla T_h\cdot n_{\Omega })}_{\partial {\Omega }_D(t)}-(\kappa \nabla w_h\cdot n_{\Omega },T_h)\partial {\Omega }_D(t)+\frac{\kappa {\gamma }_b}{h}{(w_h,T_h)}_{\partial {\Omega }_D(t)},\\ l_{bc}(w_h,T_h)& =-{(\kappa \nabla w_h\cdot n_{\Omega },T_D)}_{\partial {\Omega }_D(t)}+\frac{\kappa {\gamma }_b}{h}{(w_h,T_D)}_{\partial {\Omega }_D(t)}.\hfill \end{array}$$ (35)

The stabilization operator s_T (referred to as the ghost-penalty stabilization [3]) in (33) prevents ill-conditioning when Γ(t) approaches a node or a set of element edges. It is given by

$$s_T(w_h,T_h)≔\sum _{F\in {ℱ}_{\Gamma }(t)}{\gamma }_T\kappa h{({⟦\nabla w_h⟧}_n,{⟦\nabla T_h⟧}_n)}_F,$$ (36)

where ${⟦\nabla x⟧}_{\mathit{n}}≔{\nabla x|}_{T_F^{+}}{\mathit{n}}_F-{\nabla x|}_{T_F^{-}}{\mathit{n}}_F$ represents the jump in the normal component of ∇x over face F. The stabilization is added to a particular set of element faces ${ℱ}_{\Gamma (t)}$. Namely, these are faces of elements that intersect the interface Γ(t), excepting those exterior faces that belong to an element only in the background mesh. A typical set of ${ℱ}_{\Gamma (t)}$ is shown as the dark green edges in Fig. 3.

For the temporal discretization, a simple backward Euler scheme is implemented with time step size $\Delta t=t_{n+1}-t_n$. Given fields at time step n, the problem then becomes: find $T^{n+1}\in {𝒱}_n(t_n)$ such that

$$A_{\ast }(w,T_h^{n+1})+𝒩(w,T_h^{n+1})=L_{\ast }(w)\forall w\in {𝒱}_n(t_n),$$ (37)

where

$$\begin{array}{cc}{A}_{\ast }(w_h,T_h^{n+1})& =a_{\ast }(w_h,T_h^{n+1})+a_{bc}(w_h,T_h^{n+1})+s_T(w_h,T_h^{n+1})+{\left(P_{\theta \gamma }^{\delta }(w_h)-w_h,\kappa \nabla T_h^{n+1}\cdot {\mathit{n}}_{\Gamma }\right)}_{{\Gamma }_h(t)},\\ L_{\ast }(w_h)& =L(w_h)+\rho c_p{\left(w_h,\frac{T_h^n}{w}\right)}_{{\Omega }_h(t_n)},\hfill \\ a_{\ast }(w_h,T_h^{n+1})& =\rho c_p{\left(w_h,\frac{T_h^{n+1}}{w}\right)}_{{\Omega }_h(t_n)}+{(\nabla w_h,\kappa \nabla T_h)}_{{\Omega }_n(t_n)}.\hfill \end{array}$$ (38)

In this work, the penalty parameters are set to ${\gamma }_T={10}^{-1},\widehat{\gamma }=\gamma /h=1$, and γ_b = 100, where h is the mesh size.

3.2. Level set description of the moving interface

Recall from Section 2.4 that the motion of the boundary Γ(t) is tracked by a level-set function ϕ : Ω_b × [t₀, t_f]. We note that the level-set function is defined over the entire background domain Ω_b rather than just the active domain.

For each time step, we get $T_h^{n+1}$ from (37), thus the discrete form of (4) can be re-arranged to provide the normal velocity of the interface as

$${\mathit{v}}^{n+1}\cdot {\mathit{n}}_{\Gamma }=\frac{\kappa \nabla T^{n+1}\cdot {\mathit{n}}_{\Gamma }-I\cdot {\mathit{n}}_{\Gamma }}{\rho L_t}\text{on}\Gamma (t_n).$$ (39)

According to (39), the zero level-set contour (which describes the interface) should move with vⁿ⁺¹ · n_Γ. This is effected by first constructing an extension velocity field v_ext over the entire background domain that matches the normal velocity from (39) on the interface. The level-set function is then updated by approximating the solution to the advection equation

$$\frac{\partial \varphi }{\partial t}+{\mathit{v}}_{ext}\cdot \nabla \varphi =0$$ (40)

with the initial condition $\varphi (x,0)={\varphi }_0$, where ϕ₀ is the initial level-set function.

Defining the continuous quadratic finite element space on the background domain as

$${𝒲}_h≔\left\{v_h\in C^0({\Omega }_b)\right\},$$ (41)

the discretized advection problem becomes: find ${\varphi }_h^{n+1}\in {𝒲}_h$ such that

$$a_{\varphi }(\phi ,{\varphi }_h^{n+1})=l_{\varphi }(\phi )\forall \phi \in {𝒲}_h,$$ (42)

where

$$a_{\varphi }(\phi ,{\varphi }_h^{n+1})≔{\left(\phi +{\tau }_{SD}({\mathit{v}}_{ext}^{n+1}\cdot \nabla \phi ),\frac{{\varphi }_h^{n+1}}{\Delta t}+\theta {\mathit{v}}_{ext}^{n+1}\cdot \nabla {\varphi }_h^{n+1}\right)}_{{\Omega }_b},$$ (43)

$$l_{\varphi }(\phi )≔{\left(\phi +{\tau }_{SD}({\mathit{v}}_{\mathit{\text{ext}}}^{n+1}\cdot \nabla \phi ),\frac{{\varphi }_h^n}{\Delta t}+(1-\theta ){\mathit{v}}_{\mathit{\text{ext}}}^n\cdot \nabla {\varphi }_h^n\right)}_{{\Omega }_b},$$ (44)

with

$${\tau }_{SD}=2{\left(\frac{1}{\Delta t^2}+\frac{{\mathit{v}}_{\mathit{\text{ext}}}\cdot {\mathit{v}}_{\mathit{\text{ext}}}}{h^2}\right)}^{-\frac{1}{2}},$$ (45)

the streamline diffusion parameter [17]. In this work, the parameter θ = 0.5.

3.3. Mesh refinement strategy

In order to accurately simulate the interface evolution and the associated temperature fields, an adaptive mesh refinement strategy is adopted. The mesh intersected by the interface Γ(t) is refined at each step as shown in Figs. 3 and 4. A two-grid solution is adopted to refine and update the mesh [14,15]. First, a piecewise linear function is projected from the piece-wise quadratic zero level set function $({\varphi }_h^{n+1}=0$ in (42)) on a refined mesh. Then, this linear function is used to determine the intersection between the refined mesh and Γ. Finally, the linear approximation Γ_h and Ω_h are obtained as Fig. 4 shows.

As an alternative approach, we will also use a mesh that is selectively refined in the vicinity of the initial interface geometry, and follow the same strategy to update (cut) the mesh to obtain the linear approximation Γ_h and Ω_h.

3.4. Nonlinear solution strategy

A semi-smooth Newton–Raphson algorithm is adopted to solve (37). The Newton predictor at the (k + 1)th loop is

$$A_{\ast }(w,d{T}_h)+𝒟𝒩(w,d{T}_h;T_h^k)=r(w;T_h^k),$$ (46)

where $𝒟𝒩$ denotes the Gateaux-derivative of $𝒩$ with $d{T}_h=T_h^{k+1}-T_h^k$ such that

$$𝒟𝒩(d{T}_h,w;T_h^{\ast })≔\underset{z\to 0}{\lim }(𝒩(T_h^{\ast }+zd{T}_h,w))-𝒩(T_h^{\ast },w)),$$ (47)

where $T_h^{\ast }\in {𝒱}_h(t_h)$ is a reference temperature. In the above, the residual r of $T_h^k$ is given by

$$r(w;T_h^k)=L_{\ast }(w)-(A_{\ast }(w,T_h^k)+𝒩(w,T_h^k)).$$ (48)

A reduced space active set solver based on Newton’s method (from PETSc [2]) is adopted to enforce a lower bound constraint on T_h while solving the nonlinear system (46). This avoids the possibility of discrete temperature fields that are unrealistically small (if not negative) from forming due to the relatively large laser energy flux.

4. Experimental data and analysis

In this section, experimental results from pulsed laser ablation of BegoStone samples are reported. In addition, a relatively simple thermal energy analysis is provided to estimate what portion of the laser energy is required to generate the observed ablation of material, quantified by the volume of the damage crater in the experiment.

4.1. Experimental configuration

The experimental setup for pulsed laser ablation of wet BegoStone in air is shown in Fig. 5(a).

Fig. 5.

(a) Experimental setup for pulsed laser ablation of BegoStone samples [18], (b) schematic of the laser beam near the laser fiber tip.

A large sample of BegoStone with an initially flat surface was placed underneath an apparatus holding the laser fiber, as described in Ho et al. [18]. The fiber was oriented perpendicular to the sample surface at a short distance, on the order of a few millimeters. In what follows, this distance between the fiber tip and the originally flat surface of the sample is referred to as the standoff distance S_d.

The diameter D of the irradiated laser beam on the stone surface increases with distance from the tip of the fiber as D = D₀ + 2 tan $\theta \times \overline{S_d}(r,t)$, where D₀ denotes the core diameter of the laser fiber tip, and $\overline{S_d}$ denotes the effective standoff distance. The expansion of the beam from the tip is depicted in Fig. 5(b). In the experiments reported here, the laser fiber tip has a core diameter of D₀ = 365 μm, and θ = 15.07°. The effective standoff distance $\overline{S_d}(r,t)$ denotes the perpendicular distance from the laser fiber tip to the stone surface. We use $\overline{S_d}$ to represent the ‘original’ or ‘initial’ standoff distance, and $\overline{S_d}$ to represent the effective standoff distance which generally increases in time as the surface of the specimen is ablated.

4.2. Materials

The material used in the experiments is ‘BegoStone’ [23]. BegoStone is often employed as a “phantom” material for kidney stones, as it has acoustic and mechanical properties that are similar to human renal calculi. Prior to the laser ablation experiments, BegoStone samples (50 × 50 × 5 mm) were soaked in water for 24 h.

Although BegoStone has been mechanically characterized fairly well, its thermal properties have yet to be thoroughly measured. As a reasonable proxy, we rely on the thermal properties of gypsum. The relevant thermal properties (thermal conductivity, density, specific heat capacity, melting temperature, and latent heat of fusion) of gypsum are provided in Table 1.

Table 1.

Thermal properties of gypsum.

Property/parameter	Symbol	Value	Unit
Thermal conductivity	κ	0.17	W/(m K)
Density	ρ	2320	kg/m3
Specific heat capacity	C p	1090	J/(kg K)
Melting temperature	T m	1723.15	K
Latent heat	L t	527,500	J/kg

4.3. Experimental results

In this subsection, the results for a series of experiments where multi pulses of the laser were fired at the specimens are reported. We begin by showing a three-dimensional reconstruction of a typical crater geometry in Fig. 6. In the plot, the axes are normalized by the diameter D₀ of the laser tip and the thickness H(= 5 mm) of the BegoStone sample. In order to facilitate visualization of the changes in crater geometry for various cases, as well as comparisons to results from axisymmetric simulations, we find it useful to indicate cross-sections that happen to intersect the point of maximum depth. For example, the red point in Fig. 6 corresponds to the point of maximum depth in the crater shown. The red and blue curves are traces of the crater geometry in the x and y directions that happen to intersect this point of maximum depth.

Fig. 6.

Example of three-dimensional reconstruction of a crater geometry from a typical ablation experiment. The red and blue lines trace cross sections with the maximum depth in the x and y directions, respectively. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

4.3.1. Multi-pulse experiments

A series of multi-pulse experiments were performed for two different standoff distances of S_d = 0 mm and S_d = 0.5 mm. In the experiments, each pulse was applied to the specimen at an energy level of E_laser = 0.2 J. Laser pulses were applied at a frequency of 20 Hz and a FWHM pulse duration of t_p = 71.5 μs, for up to 1000 pulses total. Fig. 7(a)–(c) shows representative 3D and 2D cross-sectional plots for craters at various pulse numbers. The resulting volumes and areas of the observed ablated craters are plotted as a function of pulse number in Fig. 8. The data shown correspond to the mean results from a set of 10 experiments, with the error bar indicating the standard deviation.

Fig. 7.

Representative crater geometries for BegoStone samples subjected to as many as 1000 laser pulses. (a) Three-dimensional images of crater after 1000 pulses, and the cross sections after 15, 50, 75, 200, 1000 pulses of maximum depth in the (b) x and (c) y directions.

Fig. 8.

Crater (a) volume and (b) surface profile area from multi-pulse experiments on BegoStone samples for S_d = 0 and S_d = 0.5 mm.

The results generally indicate that the crater volume and areas increase quickly during the first 200 pulses, but eventually saturate at some point. Finally, we note that the volumes and areas decrease as the standoff distance is increased. This is due to the fact that the energy density of the laser impacting the sample at any given point on the surface decreases as the distance between the surface and the laser tip increases.

4.4. Energetic analysis

We now perform a simple theoretical analysis to estimate what portion of the total laser energy delivered to the samples is used to ablate the material. Considering the total ablated volume for a BegoStone specimen, one can calculate the energy required to first raise the temperature of the entire ablated volume uniformly to the melting temperature, and then surpass the latent heat. Of course, in the actual experiments, portions of the ablated material reach different temperatures at different times. But as a rough estimate it is useful to assume that everything is heated uniformly, together. The ratio of this energy level to the total amount of energy that is supplied by the laser then provides some insight into the percentage that is actually being used for ablation.

The total energy Q required for ablation of a volume of material V_a can be calculated by

$$\begin{array}{cc}Q& =Q_1+Q_2,\\ Q_1& =\rho V_a{c}_p\Delta T,\\ Q_2& =\rho V_a{L}_t,\hfill \end{array}$$ (49)

where Q₁ and Q₂ stand for the energy required for the initial temperature increase and the phase transformation, respectively. Thus the “effective absorption ratio” of the laser energy can be calculated

$$\overline{a}=\frac{Q}{E_{\text{laser }}}=\frac{Q_1+Q_2}{E_{\text{laser }}}=\frac{\rho V_a{c}_p\Delta T+\rho V_a{L}_t}{E_{\text{laser }}}.$$ (50)

Performing the energy analysis for the multi-pulse experiments gives the effective absorption ratios shown in Fig. 9. The ratios are calculated at different pulse numbers. We note that for the multi-pulse experiments E_laser = (0.2 × n) J, where n denotes the total number of pulses. The results indicate that, at early times, approximately 2% of the laser energy is required to ablate the material. This percentage decreases rapidly over the first 200 pulses, before saturating. The saturation corresponds to the results shown in Fig. 8, where at some point material ceases to be removed with subsequent laser pulses.

Fig. 9.

Effective absorption ratios ā for multi-pulse experiments on BegoStone specimens, for two different standoff distances.

From the energetic analysis of the laser ablation experiments, it is clear that only a relatively small portion of the total laser energy is likely needed to ablate the material. The magnitudes of effective absorption seems to be very small for the multi-pulse experiments, where the maximum percentage does not even reach 2%. These results indicate that in order for a numerical model based only on energetics to be calibrated against ablation experiments, the amount of laser energy the model assumes to be absorbed at the surface likely needs to be reduced compared to what is applied. We will examine this in the numerical results that follow in the next section.

5. Numerical simulations

In this section, we first calibrate our axisymmetric model against the experimental results. Then a series of simulations are performed with the calibrated model. The simulation results are shown and compared to the experiments to show the ability of the proposed model and provide some insights into the physics of laser lithotripsy.

5.1. Calibration of the 2D axisymmetric model

Recall the laser beam profile as shown in Fig. 5(b), where the diameter D of the beam expands with distance $\overline{S_d}$ from the tip as D = D₀ + 2 tan $\theta \times \overline{S_d}(r,t)$. Consistent with the observations in the previous section, the amplitude of the laser is adjusted by a calibration constant c₁ that reflects the fact that only a portion of the incident energy is actually absorbed by the material. Fig. 10 illustrates the resulting laser energy fluence profile c₁ · I (W/mm³ ) using an absorption c₁ = 2% of the total energy, and a standoff distance of S_d = 0 mm.

Fig. 10.

Laser energy fluence: (a) total (b) zoom for [0, 2.7]x/D₀ × [0, 1.01]z/H, using E_laser = 0.2 J and absorption coefficient c₁ = 0.02.

To begin, we consider a computational domain defined as Ω_b = [0, 13.7]D₀ × [0, 1.01]H = 5 × 5 mm, where θ = 15.07°, D₀ = 0.365 mm represents the initial diameter of the laser beam, and H = 5 mm is the height of the BegoStone specimen. Although this corresponds to a cylindrical geometry (and not a block) in an axisymmetric setting, the right side of the virtual cylinder is sufficiently far from where the crater forms that boundary effects are assumed negligible. The initial zero level set (the initial interface/top surface) is set to ϕ(x, 0) = y − H. The initial temperature field is set to T = 20 °C, consistent with room temperature in the experiments. Zero-flux boundary conditions are prescribed on all surfaces with the exception of the top, where the laser energy flux I is applied to the evolving surface Γ. A physically-reasonable alternative to the boundary conditions on the right and bottom surfaces could have been to prescribe them to be fixed at room temperature, but this would likely give rise to more conservative temperature profiles than expected. Regardless, given the overall size of the computational domain it is unlikely that the far field conditions significantly impact the results.

In the simulations, three different meshes (of N × N elements) are used. To aid in visualization, the computational domain is mirrored about the y-axis in the figures. The mesh of the physical geometry (that remains after ablation) is shown, as well as contours from experimental measurements corresponding to the maximum crater depth (as indicated in Fig. 6). The results indicate that once c₁ is calibrated to match the total volume, the simulated crater geometry matches the experimental measurements reasonably well. To ensure the h-refinement of the method, a single pulse experiment and simulation are performed for the specific laser setting (E_laser = 1.2 J, t_p = 179.5 μs, and c₁ = 0.185). Fig. 11 shows the h-refinement results of these meshes, the results indicate that the model-based simulation results are spatially converged. In the following multi-pulse simulations, mesh of N = 60 is used.

Fig. 11.

Comparison of cross-section ablated surface geometry (in cyan) and experimental measurements (in red and blue) for a sequence of increasingly refined meshes (with N indicating refinement level). Results are shown for a laser energy of E_laser = 1.2 J, t_p = 179.5 μs, and an absorption fraction of c₁ = 0.185. Only a portion of the total computational domain is shown, to focus attention on the crater region. The red and blue lines trace cross sections through a point of maximum depth in the x and y directions from the experiment. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

5.2. Multi-pulse simulations

For multi-pulse laser ablation, simulations are performed corresponding to multiple pulses at E = 0.2 J, a typical clinical setting for stone dusting treatment in laser lithotripsy [5]. Returning to the analysis in Section 4.4, we recall that the experimental results indicate that at most 2% of the incoming laser energy is required to ablate the material on the surface. Accordingly, in what follows, we report results corresponding to ablation ratios of 0.5%, 1%, and 2%. To accelerate the multi-pulse thermal damage simulations, a frequency of 5000 Hz is simulated. The actual experiments were conducted at a frequency of 20 Hz, which is obviously quite slower. In our experience with our model-based simulations, we did not observe a large difference when the simulated frequency was decreased below 5000 Hz. Lower frequencies simply correspond to longer periods of time when the laser is idling and not delivering energy to the surface of the specimen. As a result, the simulation results at the higher frequency serve to bound the experimental measurements, as the simulation results effectively correspond to a greater amount of energy delivery to the virtual specimen per unit time.

Figure 12 compares the simulation results to the experimental measurements for a standoff distance of S_d = 0 mm. Simulations were performed for only 500 pulses, as this was the point when all crater volumes had saturated. The results indicate that absorption ratios of 0.5% and 2% effectively bound the experimental measurements. In terms of matching the experimental measurements for the crater volume and area, an absorption ratio of 1% appears to provide a reasonably good correspondence.

Fig. 12.

Crater (a) volume and (b) surface profile area from model-based simulations for 0.2 J, 500 pulses, S_d = 0 mm, and varying absorption percentages.

The thermal ablation simulation results for 0.2 J, 1% absorption and varying S_d for multiple laser pulses are shown in Fig. 13. The comparison between the experimental observations and the simulation results once again indicate that c₁ = 1% model tracks the real ablation behavior well for both S_d = 0 and S_d = 0.5 mm cases.

Fig. 13.

(a) Crater volume and (b) area from model-based simulations for 0.2 J, 500 pulses, 1% absorption, and varying S_d.

We note that both the simulation and the experimental results for the crater volumes saturate around 200 pulses. This outcome is easily explained by the model-based simulation. In essence, as the surface ablates, the distance between the surface and the tip of the laser increases. Consistent with the spreading of the laser beam with increased standoff distance, points further away from the laser tip experience lower energy density than points that are closer. As such, after a sufficient level of ablation has occurred, the surface is too far from the laser beam to become sufficiently heated and reach the melting temperature. This phenomena also helps explain why the crater volumes and areas decrease when the standoff distance is increased. As shown in Fig. 13(b), the areas also saturate earlier (at around 100 pulses) for a standoff distance of S_d = 0.5 mm compared to S_d = 0 cases (at around 200 pulses).

The geometry of a typical experimental crater at 200-pulses and a standoff distance of S_d = 0 mm is compared against simulation results (for various absorption ratios) in Fig. 14. These results also indicate that an absorption ratio of 1% yields the best comparison between the simulation results and the experiments. In Fig. 15, the crater geometries from multi pulse simulations are plotted and compared to the experiments, using c₁ = 0.01.

Fig. 14.

Crater geometries after 200 pulses of simulations and experiments, (a)-(c): c₁ = 0.5%, 1%, 2%.

Fig. 15.

Crater geometries of simulations (in cyan) and experiments (in red and blue) for BegoStone samples, corresponding to different number of pulses and standoff distances, and an absorption fraction of c₁ = 0.01. The red and blue lines trace cross sections with maximum depth in the x and y directions from the experiments. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

The evaluated effective absorption ā for multi-pulse experiments and simulations for varying absorption and standoff distances are illustrated in Fig. 16. To calculate ā for the simulation results, we use (50) but with the calculated volumes instead of the experimentally measured volumes. In comparison to the experimental data, the effective absorption reveals that the simulation tracks the changing trends of the crater volume as the ablation proceeds. From Fig. 16, we note that the subtle difference between the experimental data of varying standoff distances is also captured. This shows that the proposed energetic model, while relatively simple, nevertheless captures important aspects of the underlying physics of the laser ablation process.

Fig. 16.

Effective absorption ratios ā for multi-pulse experiments and simulations on BegoStone specimens, for (a) varying absorption, S_d = 0 mm, (b) two different standoff distances at 1% absorption.

In the current work, the use of an effective absorption constant ā provides a means for a relatively simple model of ablation to nevertheless provide a good match with experimental observations. This parameter essentially approximates the loss of useful laser energy associated with many physical processes that are not explicitly accounted for in the model. These include, for example, laser scattering, the actual physical absorptivity of the surface, coupled thermo-mechanical-chemical effects, and etc. As a result, however, it provides insight into the percentage of laser energy that is being used for ablation and also helps explain why ablation saturates with increasing pulses.

6. Summary and concluding remarks

In the present study, a model of thermal ablation and an accompanying embedded finite element method are modified to simulate recent experiments of pulsed laser ablation of BegoStone samples. In Section 2, the method of Claus et al. [8] is reformulated for axisymmetric problems, where the level set method is used to track the interface, and Nitsche’s method is adopted to impose the Signorini condition on the moving interface. In Section 3, the discretization of the modified embedded finite element method is presented with a mesh refinement strategy and a nonlinear solution strategy. In Section 4, results from both single shot and multi-pulsed laser ablation experiments conducted on BegoStone samples are reported. Additionally, a straightforward energetic analysis is performed to estimate the percentage of incoming laser energy that is required to ablate the observed volumes. In Section 5, the axisymmetric model is calibrated against experimental measurements of laser ablation experiments. In particular, the model is calibrated against the experimental measurements for ablated volumes by adjusting the absorption percentage. Then, simulation results are validated against observations of multi-pulsed experiments for the crater volume, area, and geometry as a function of laser pulse energy and duration.

The calibrated model is seen to yield reasonable results (crater volume, area and geometry) compared to multi-pulsed laser ablation experiments. The crater volumes increase during the first 200 pulses and then saturate after approximately 400 pulses. Our results show that the spreading of the laser beam from the fiber tip is necessary and needs to be included in the model in order to obtain a qualitative match with the experimental data. As the experimental data and model-based simulations proceed, the crater forms and the distance between the laser beam tip and the top surface of the residual stone material increases, and the laser fluence and irradiance reach to the residual stone surface become more diffused. As a result, the crater volume will eventually saturate once the ablated surface moves too far away from the laser beam for the incoming energy to heat the material above its melting temperature. The observation that only about 1% of the laser pulse energy is used to create thermal ablation also supports recent studies demonstrating that other mechanisms, such as cavitation erosion, may play a critical role in stone dusting during laser lithotripsy [5].

The model-based simulation results and their comparison to the experimental data also suggest a number of areas for future research. These include incorporating additional physical aspects of ablation into the model, such as thermo-mechanical effects, laser scattering, and fluid flow in the melt pool. The experimental data also indicate a lack of spatial uniformity in the response, which might be due to variations in material properties and scattering of the laser pulses by damaged particles (dust) removed from the stone surface. The model results could be enhanced by incorporating such stochastic elements and conducting fully threedimensional simulations.

Data Availability

Data will be made available on request.