Techniques for element formulation and quadtree-based triangular mesh generation for strain-based finite elements

Abstract

Finite elements are often formulated by imposing sufficient conditions to ensure convergence and good accuracy. This work demonstrates a new technique to impose compatibility and equilibrium conditions for membrane finite elements that are formulated based on the strain approach.

• •The compatibility and equilibrium conditions are imposed onto the initial formulations (or test functions) by using corrective coefficients (c₁, c₂, and c₃).
• •The technique is found to be capable of producing alternate or similar forms for the test functions. Performances of the resultant (or final) formulations are shown by solving three benchmark problems.

Additionally, a new technique to formulate strain-based triangular transition elements (denoted as SB-TTE) is introduced.

• •The new technique introduces another node (the fourth node) at one of the sides of a strain-based triangular element (mid-node, which is needed for the quadtree-based triangular mesh generation) without adding a degree of freedom.

Graphical abstract

Specifications table

Subject area	Engineering
More specific subject area:	Finite element method
Name of your method:	•Use of corrective coefficients for the formulation of strain-based finite elements. •Strain-based triangular transition element (SB-TTE)
Name and reference of original method:	1.A. B. Sabir, “A rectangular and triangular plane elasticity element with drilling degrees of freedom,” in Proceeding of the 2nd International conference on variational methods in engineering, Berlin, 1985. URL:https://books.google.com.my/books/about/Variational_Methods_in_Engineering.html?id=cfp8AAAAIAAJ&redir_esc=y 2.A. I. Mousa and S. M. Tayeh, “New Strain-Based Triangular and Rectangular Finite Elements for Plane Elasticity Problems,” M. S. Thesis, Faculty of Engineering, The Islamic University of Gaza, 2003. URL: https://library.iugaza.edu.ps/Thesis/53850.pdf 3.M. T. Belarbi and M. Bourezane, “An assumed strain based on triangular element with drilling rotation,” Courrier du Savoir scientifique et technique, vol. 6, no. 6, Pages 117–123, 2005. URL: http://www.webreview.dz/IMG/pdf/18-Bouraz.pdf 4.X. H. Tang, S. C. Wu, C. Zheng and J. H. Zhang, “A novel virtual node method for polygonal elements,” Applied Mathematics and Mechanics, vol. 30, no. 10, pp. 1233–1246, 2009. DOI: https://doi.org/10.1007/s10483–009–1003–3 5.L. Perumal, “A novel virtual node hexahedral element with exact integration and octree meshing,” Mathematical Problems in Engineering, vol. 2016, article ID: 3261391, pp. 1-19, 2016. DOI: https://doi.org/10.1155/2016/3261391
Resource availability:	The methods are explained in this paper.

Method details

Introduction

Recently, strain-based finite elements have been revisited, and many works have been performed on the element formulations, including extending the method to three dimensions. Strain-based elements have several advantages as compared to displacement-based elements [1]. So far, strain-based finite elements have been developed to solve problems in solid mechanics. For example, strain-based elements have been used in the analysis of plates (subjected to static and free vibrations as well as bending) [2], [3], [4], shell structures [5], [6], [7], cylindrical panels [8], circular structures [9], and plane elasticity (static/linear and dynamic analyses including vibrations) [10], [11], [12], [13], [14], [15], [16].

The principle formulation of strain-based elements starts with expressing the strain variables (instead of displacements) using suitable functions. This is because the strain-based approach prioritizes fulfillment of conditions that are associated to strain components rather than the displacements. Therefore, to ensure convergence and reasonable accuracy of the results, functions representing the variation of the strain variables within the element should satisfy the compatibility and equilibrium conditions. The compatibility equation for strain-based elements (for 2-dimensional solid mechanics problems) is given as follows:

$$\frac{{\partial }^2{\epsilon }_x}{\partial y^2}+\frac{{\partial }^2{\epsilon }_y}{\partial x^2}-\frac{{\partial }^2{\gamma }_{xy}}{\partial x\partial y}=0$$ (1)

Normal strains in x and y directions are represented by the symbols ${\epsilon }_x$ and ${\epsilon }_y$, respectively, while the shear strain is represented by the symbol ${\gamma }_{xy}$ in the Eq. (1). Equilibrium conditions should also be met to represent the structure's static equilibrium or uniform motion. Equilibrium is attained when the sum of the internal forces at the element nodes equals the sum of the external loads that act on the structure. Satisfaction of the equilibrium conditions is considered imperative in structural analysis and design [17]. Equilibrium equations for strain-based elements (for 2-dimensional solid mechanics problems) are given in terms of the stress components as follows:

$$\frac{\partial {\sigma }_x}{\partial x}+\frac{\partial {\tau }_{xy}}{\partial y}=0$$ (2)

$$\frac{\partial {\sigma }_y}{\partial y}+\frac{\partial {\tau }_{xy}}{\partial x}=0$$ (3)

${\sigma }_x$ and ${\sigma }_y$ represent the normal stresses in x and y directions, respectively. Shear stress is represented by the symbol ${\tau }_{xy}$. The stresses in terms of the strains, shear modulus of elasticity, G and Lame constant, $\lambda$, are given as follows:

$${\sigma }_x=2G{\epsilon }_x+\lambda \left({\epsilon }_x+{\epsilon }_y\right)$$ (4)

$${\sigma }_y=2G{\epsilon }_y+\lambda \left({\epsilon }_x+{\epsilon }_y\right)$$ (5)

$${\tau }_{xy}=G{\gamma }_{xy}$$ (6)

where

$$G=\frac{E}{2\left(1+v\right)}$$ (7)

$$\lambda =\frac{vE}{\left(1+v\right)\left(1-v\right)}\left(\text{for}\text{plane}\text{stress}\right)$$ (8)

$$\lambda =\frac{vE}{\left(1+v\right)\left(1-2v\right)}\left(\text{for}\text{plane}\text{strain}\right)$$ (9)

E and v are Young's modulus of elasticity and Poisson ratio, respectively. The early strain-based finite element formulations are obtained through trial and error by modifying the original functions proposed by Sabir [18] or by other means. These early formulations satisfy the compatibility condition but do not satisfy the equilibrium equations. For example, when the assumed functions for the strain variables from Ref. [10] are put into the compatibility equation Eq. (1)), a value of zero is obtained (right-hand side of Eq. (1)), indicating that compatibility is satisfied. However, when the same functions for the strain variables are put into Eqs. (2) and ((3) and solved, zero values are not obtained, signifying that the equilibrium conditions are not met. Formulations that do not satisfy both conditions are prevalent in the literature [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16],18]. Therefore, the application of these formulations is restricted [19], [20], [21]. However, it can be seen that some of the strain-based elements that satisfy only the compatibility conditions (do not satisfy equilibrium conditions) are still able to provide converging and accurate results. Recent works on incorporating strain-based finite elements with existing plate theories (such as Mindlin and Kirchhoff plate theories) for the analysis of plates can be seen in studies by Boussem et al. [22] and Belounar et al. [23].

In short, many researchers have presented new formulations for the strain-based elements that satisfy both compatibility and equilibrium conditions [20], [21],[24], [25], [26], [27], [28], [29], [30], [31], [32], [33]. However, only a handful of them (such as [20], [21],25,33]) have described the techniques used to obtain the functions that satisfy both conditions, causing the techniques used in the remainder to be left unknown. This paper introduces two new techniques that are applicable to strain-based finite elements. First technique is a method to impose both compatibility and equilibrium conditions into the initial formulations by using corrective coefficients, C₁, C₂, and C₃, while the second is a new quadtree mesh generation technique.

This paper is arranged as follows. Background of strain-based finite elements is provided in Section 2. The latter part of Section 2 shows finite element equations for a strain-based element in terms of shape functions (alternate approach). Techniques to impose both compatibility and equilibrium conditions for element formulation are described in Section 3. This section also includes simulation results and a summary (as method validation). Section 4 shows development of the strain-based triangular transition element (SB-TTE) and implementation of these elements in quadtree-based triangular mesh generation. The paper is finally concluded in Section 5.

Background of strain-based finite element equations

The principle formulation of strain-based membrane element starts with expressing the strain variables (ε_x, ε_y, and γ_xy) by using suitable functions, f in the form:

$${\epsilon }_x,{\epsilon }_y,{\gamma }_{xy}=f\left(x,y,a_i\right)$$ (10)

Usually, polynomial functions are used to represent these strain variables, and the polynomial terms are selected from the Pascal triangle. Degrees of freedom, a_i (i = 1, 2, …, n), are then distributed among these strain variables. Total number of coefficients, a_i, is equal to the total number of degrees of freedom for the entire element, n. Expressions for the displacements (U and V) are then obtained by using the strain-displacement relations. The strain-displacement relations are given as follows:

$${\epsilon }_x=\frac{\partial U}{\partial x}$$ (11)

$${\epsilon }_y=\frac{\partial V}{\partial y}$$ (12)

$${\gamma }_{xy}=\frac{\partial U}{\partial y}+\frac{\partial V}{\partial x}$$ (13)

Strain-based finite elements are formulated by incorporating the rigid body and straining modes. Rigid body mode is imposed by letting the strain components in Eqs. (11) to (13) to be zero, and the corresponding functions for the displacements are obtained through integration. This results in permanent assignment of the first three degrees of freedom (a₁, a₂, and a₃) to the displacements $U_{rigid}$ and $V_{rigid}$ [11]:

$$U_{rigid}=a_1-a_{3}y$$ (14)

$$V_{rigid}=a_2+a_{3}x$$ (15)

Remaining degrees of freedom are then used to represent the straining mode of the element. Unlike the rigid body mode, assignment of the degrees of freedom and the polynomial terms for the straining mode is not permanent; therefore, many possible combinations exist. Examples of functions that are used to represent the straining mode are shown here, which were proposed by Sabir [34]:

$${\epsilon }_x=a_4+a_{5}y+a_{7}x$$ (16)

$${\epsilon }_y=a_6+a_{7}x+a_{5}y$$ (17)

$${\gamma }_{xy}=a_8+a_9\left(x+y\right)$$ (18)

The strain-displacement relations Eqs. (11) to ((13)) can then be used to obtain the displacement functions for the straining mode, $U_{strain}$ and $V_{strain}$. Initial functions for straining displacements ($U_{strain}$ and $V_{strain}$) are obtained by substituting functions for the normal components of the strain (from Eqs (16) to (18)) into strain displacement relationship formulas Eqs (11) to ((13)) and performing the integrations. These initial straining displacements are in terms of unknown functions that resulted from the integrations. These unknown functions are solved by comparison of two similar expressions for the shear strain, ${\gamma }_{xy}$. The first expression for ${\gamma }_{xy}$ is obtained by substituting the functions for the initial straining displacements into Eq. (13), while the second expression is provided by Eq. (18). The final expressions for displacements U and V are then obtained by summing the rigid body and straining modes together:

$$U=U_{rigid}+U_{strain}$$ (19)

$$V=V_{rigid}+V_{strain}$$ (20)

Apart from the displacements, the rotational degree of freedom, θ, can also be introduced to improve the accuracy of the solutions, particularly when bending is involved. This is because introduction of rotational degree of freedom to the straining mode makes the element matrix to be less stiff to deformation. Rotational degree of freedom is obtained by using the following equation:

$$\theta =\frac{1}{2}\left(\frac{\partial V}{\partial x}-\frac{\partial U}{\partial y}\right)$$ (21)

The field variables (displacements and rotational degree of freedom) can be represented in matrix form as:

$$\left\{q\right\}=\left[p\right]\left\{a\right\}$$ (22)

where $\left\{q\right\}={\left\{U,V,\theta \right\}}^T$ is the field variable matrix, $\left[p\right]$ is the field variable interpolation matrix, and $\left\{a\right\}$ is the matrix containing all the degrees of freedom. The nodal field variable interpolation matrix, [P] for a triangular element with three nodes is then given by:

$$\left[P\right]={\left[p_i{p}_j{p}_k\right]}^T$$ (23)

where $p_i=p\left(x_i,y_i\right)$, $p_j=p\left(x_j,y_j\right)$ and $p_k=p\left(x_k,y_k\right)$ are the field variable interpolation matrices for each node of a triangular element, respectively. Nodal field variable matrix for the element (with three degrees of freedom for each node) is:

$$\left\{Q\right\}={\left\{U_1,V_1,{\theta }_1,U_2,V_2,{\theta }_2,U_3,V_3,{\theta }_3\right\}}^T$$ (24)

Strain interpolation matrix, [B], contains the polynomial terms and constants that are used for the formulation (definition of variation of the strain components within the element). The [B] matrix for the formulation in Eqs. (16) to (18) is:

$$\left[B\right]=\left[\begin{array}{cc}0& 0\\ 0& 0\\ 0& 0\end{array}\begin{array}{cc}0& 1\\ 0& 0\\ 0& 0\end{array}\begin{array}{cc}y& 0\\ y& 1\\ 0& 0\end{array}\begin{array}{cc}x& 0\\ x& 0\\ 0& 1\end{array}\begin{array}{c}0\\ 0\\ \left(x+y\right)\end{array}\right]$$ (25)

The stiffness matrix is obtained by using the formula:

$$\left[K_e\right]={\left[P^{-1}\right]}^T\left[\int \underset{\Omega }{\overset{.}{\int }}\left({\left[B\right]}^T\left[D\right]\left[B\right]\right)d\Omega \right]\left[P^{-1}\right]$$ (26)

where D is the constitutive matrix and $\Omega$ is the element domain.

Strain-based finite element equations in terms of shape functions

The finite element equations provided in Section 2.0 can be represented in terms of shape functions, N. The field variable matrix, $\left\{q\right\}$ given by Eq. (22) can be written in terms of field variable interpolation matrix $\left[p\right]$, nodal field variable interpolation matrix, [P] and nodal field variable matrix $\left\{Q\right\}$, as follows:

$$\left\{q\right\}=\left[p\right]{\left[P\right]}^{-1}\left\{Q\right\}$$ (27)

Product of $\left[p\right]{\left[P\right]}^{-1}$ represents the matrix containing the shape functions, $\left[N\right]$:

$$\left\{q\right\}=\left[N\right]\left\{Q\right\}$$ (28)

where $\left[N\right]={\left[\left\{N^U\right\},\left\{N^V\right\},\left\{N^{\theta }\right\}\right]}^T$. For a three nodes triangular element with three degrees of freedom per node (n = 9), the shape functions are given as:

$$\left\{N^U\right\}=\left\{N_1^U,N_2^U,\dots ,N_n^U\right\}$$ (29)

$$\left\{N^V\right\}=\left\{N_1^V,N_2^V,\dots ,N_n^V\right\}$$ (30)

$$\left\{N^{\theta }\right\}=\left\{N_1^{\theta },N_2^{\theta },\dots ,N_n^{\theta }\right\}$$ (31)

Eqs. (28) to (31) and (24) show that, unlike the conventional displacement-based elements, calculation of a particular field variable for the strain-based element is dependent on all the other field variables as well. The strain interpolation matrix, B, is then given by:

$$\left[B\right]=\left[\begin{array}{c}\frac{\partial \left\{N^U\right\}}{\partial x}\\ \frac{\partial \left\{N^V\right\}}{\partial y}\\ \frac{\partial \left\{N^U\right\}}{\partial y}+\frac{\partial \left\{N^V\right\}}{\partial x}\end{array}\right]$$ (32)

Element stiffness matrix can be obtained by using the formula:

$$\left[K_e\right]=\left[\int \underset{\Omega }{\overset{.}{\int }}\left({\left[B\right]}^T\left[D\right]\left[B\right]\right)d\Omega \right]$$ (33)

Both approaches (equations in Sections 2 and 2.1) yield the same results. However, the second approach (representation of the finite element equations by using shape functions) can be used to formulate transition elements that are required for quadtree-based triangular mesh generation. Techniques to formulate a transition element and quadtree-based triangular mesh are described in Section 4. Upcoming Section 3 describes the method to formulate strain-based finite elements that satisfy compatibility and equilibrium conditions.

The method

Compatibility and equilibrium conditions can be imposed onto the functions representing the straining mode by using corrective coefficients (c₁, c₂, and c₃). The procedures are:

• 1.Introduce corrective coefficients (c₁, c₂ and c₃) to the initial strain functions (test functions) representing the straining mode as new separate terms. These coefficients are used to enforce the compatibility relation given by Eq. (1) and equilibrium conditions given by Eqs. (2) and (3). A total of three corrective coefficients is used since there are three equations or conditions to be satisfied.
• 2.These corrective coefficients are accompanied by imaginary or temporary degrees of freedom (a_c1, a_c2, and a_c3, respectively) and polynomial terms from the Pascal triangle.
• 3.Polynomial terms from the Pascal triangle that accompany the corrective coefficients are selected so that overall terms within the function for a particular strain component become complete or symmetrical.
• 4.The expressions that resulted from Steps 1–3 above are substituted into Eqs. (1) to (3). These equations are then solved simultaneously for the corrective coefficients (c₁, c₂, and c₃), which will be expressed in terms of a_c1, a_c2, and a_c3 and polynomial terms from the Pascal triangle.
• 5.These corrective coefficients are then substituted into the formulations and checked to see whether the compatibility and equilibrium conditions are fulfilled.

The method is implemented into several test functions (TFs). Resultant functions (RFs) for the straining mode after incorporating both compatibility and equilibrium conditions by using the proposed technique are provided in Table 1. There are two approaches to solve for the corrective coefficients. The first approach treats the corrective coefficients as functions of x and y, while the second approach treats them as constants. Results for both approaches and different assignments of the corrective coefficients are given in Table 1. TF₁ and TF₂ are existing formulations from the literature mentioned in Table 1, while TF₃ and TF₄ are new. TF₃ is formed by ensuring that all complete and symmetrical polynomial terms from the Pascal triangle are assigned to each strain component individually, but with a different form than TF₁. Moreover, sharing of degrees of freedom among the strain components in TF₃ is lesser than in TF₁ (a₅ and a₇ are shared among the normal strain components of TF₁, while TF₃ shares only a₈). The last test function, TF₄, is formed by slightly modifying TF₃ by adding one degree of freedom (a₁₀) to avoid sharing of degrees of freedom among the strain components.

Table 1.

Test functions (TFs) and corresponding resultant functions (RFs).

Test Function (TF)	Strain functions for straining mode	Assignment of corrective coefficients	Resultant functions (RFs) for straining mode (satisfy both equilibrium and compatibility conditions)
			c1, c2,= f (x, y), c3=0 (Approach 1)	c1, c2,c3 = constants (Approach 2)
TF1[34]	${\epsilon }_x=a_4+a_{5}y+a_{7}x$ ${\epsilon }_y=a_6+a_{7}x+a_{5}y$ ${\gamma }_{xy}=a_8+a_9\left(x+y\right)$	a) ${\epsilon }_x=a_4+a_{5}y+a_{7}x+c_1{a}_{c1}xy$ ${\epsilon }_y=a_6+a_{7}x+a_{5}y+c_2{a}_{c2}xy$ ${\gamma }_{xy}=a_8+a_9\left(x+y\right)+c_3{a}_{c3}xy$	${\epsilon }_x=a_4+\left(\frac{y}{1-v}\right)a_5+\left(\frac{vx}{v-1}\right)a_7+\left(\frac{vy-x}{2\left(v+1\right)}\right)a_9$ ${\epsilon }_y=\left(\frac{vy}{v-1}\right)a_5+a_6+\left(\frac{x}{1-v}\right)a_7+\left(\frac{vx-y}{2\left(v+1\right)}\right)a_9$ ${\gamma }_{xy}=a_8+a_9\left(x+y\right)$	Same as approach 1
		b) ${\epsilon }_x=a_4+a_{5}y+c_1{a}_{7}x$ ${\epsilon }_y=a_6+a_{7}x++c_2{a}_{5}y$${\gamma }_{xy}=a_8+a_9\left(x+y\right)+c_3{a}_{c3}xy$	${\epsilon }_x=a_4+y{a}_5-\left(vx\right)a_7+\left(\frac{v-1}{2}\right)x{a}_9$ ${\epsilon }_y=-\left(vy\right)a_5+a_6+x{a}_7+\left(\frac{v-1}{2}\right)y{a}_9$ ${\gamma }_{xy}=a_8+a_9\left(x+y\right)$	Same as approach 1
TF2[12]	${\epsilon }_x=a_4+a_{5}y$ ${\epsilon }_y=a_6+a_{7}x$ ${\gamma }_{xy}=a_8+a_9\left(x^2+y^2\right)$	a) ${\epsilon }_x=a_4+a_{5}y+c_1{a}_{c1}x$ ${\epsilon }_y=a_6+a_{7}x+c_2{a}_{c2}y$${\gamma }_{xy}=a_8+a_9\left(x^2+y^2\right)+c_3{a}_{c3}xy$	${\epsilon }_x=a_4+\left(\frac{y}{1-v^2}\right)a_5+\left(\frac{vx}{v^2-1}\right)a_7+\left(\frac{v-1}{v+1}\right)xy{a}_9$ ${\epsilon }_y=\left(\frac{vy}{v^2-1}\right)a_5+a_6+\left(\frac{x}{1-v^2}\right)a_7+\left(\frac{v-1}{v+1}\right)xy{a}_9$${\gamma }_{xy}=a_8+a_9\left(x^2+y^2\right)$	Not available
		b) ${\epsilon }_x=a_4+c_1{a}_{5}y$ ${\epsilon }_y=a_6+c_2{a}_{7}x$ ${\gamma }_{xy}=a_8+a_9\left(x^2+y^2\right)+c_3{a}_{c3}xy$	${\epsilon }_x=a_4+\left(\frac{v-1}{v+1}\right)xy{a}_9$ ${\epsilon }_y=a_6+\left(\frac{v-1}{v+1}\right)xy{a}_9$ ${\gamma }_{xy}=a_8+a_9\left(x^2+y^2\right)$	Not available
TF3	${\epsilon }_x=a_4+a_{7}x+a_{8}y$ ${\epsilon }_y=a_5+a_{8}x+a_{9}y$ ${\gamma }_{xy}=a_6$	${\epsilon }_x=a_4+a_{7}x+a_{8}y$ ${\epsilon }_y=a_5+a_{8}x+a_{9}y$ ${\gamma }_{xy}=a_6+c_1{a}_{c1}x+c_2{a}_{c2}y$	Not available	${\epsilon }_x=a_4+a_{7}x+a_{8}y$ ${\epsilon }_y=a_5+a_{8}x+a_{9}y$ ${\gamma }_{xy}=a_6\left(\frac{2y}{v-1}\right)a_7+\left(\frac{2v}{v-1}\right)\left(x+y\right)a_8+\left(\frac{2x}{v-1}\right)a_9$
TF4	${\epsilon }_x=a_4+a_{7}x+a_{8}y$ ${\epsilon }_y=a_5+a_{9}x+a_{10}y$ ${\gamma }_{xy}=a_6$	${\epsilon }_x=a_4+a_{7}x+a_{8}y$ ${\epsilon }_y=a_5+a_{9}x+a_{10}y$ ${\gamma }_{xy}=a_6+c_1{a}_{c1}x+c_2{a}_{c2}y$	Not available	${\epsilon }_x=a_4+a_{7}x+a_{8}y$ ${\epsilon }_y=a_5+a_{9}x+a_{10}y$ ${\gamma }_{xy}=a_6+\left(\frac{2y}{v-1}\right)a_7+\left(\frac{2v}{v-1}\right)\left(x\right)a_8+\left(\frac{2v}{v-1}\right)\left(y\right)a_9+\left(\frac{2x}{v-1}\right)a_{10}$

Assignment of all three corrective coefficients among the three strain components and solving using the first approach leads to simultaneous equations with the same binomial expressions with insufficient boundary conditions. This can be avoided by using only two corrective coefficients that are distributed among two of the strain components but may cause loss of one of the requirements (especially when higher order terms are present). Assignment of both corrective coefficients to a single strain component is also found to lead to unsolvable simultaneous equations. These problems can be avoided when the second approach is used (as shown by TF₃ and TF₄), but it does not guarantee solutions (that meet both conditions) for all the cases, such as TF₂. Assigning the corrective coefficients to existing degrees of freedom (instead of the imaginary or temporary degrees of freedom) causes the particular degrees of freedom to vanish and leads to singular matrices, as shown by TF₂(b). Assignment of the corrective coefficients to existing degrees of freedom can be done if similar degrees of freedom are used in the functions for the other strain components, as verified by TF₁(b).

Application of the method for TF₁(a) is described below for Approach 1, followed by Approach 2. The test functions after assignment of the corrective coefficients (TF₁(a)) are:

$${\epsilon }_x=a_4+a_{5}y+a_{7}x+c_1{a}_{c1}xy$$ (34)

$${\epsilon }_y=a_6+a_{7}x+a_{5}y+c_2{a}_{c2}xy$$ (35)

$${\gamma }_{xy}=a_8+a_9\left(x+y\right)+c_3{a}_{c3}xy$$ (36)

Only two corrective coefficients are used for Approach 1 ($c_3=0$) to avoid simultaneous equations with the same binomial expressions with insufficient boundary conditions. The compatibility condition given by Eq. (1) is met since all three functions consist of first-order polynomial terms. Substituting Eqs. (34) to (36) above into Eqs. (2) and (3) yields:

$$2G\left(a_7+\frac{\partial \left(c_1{a}_{c1}xy\right)}{\partial x}\right)+\lambda \left(a_7+\frac{\partial \left(c_1{a}_{c1}xy\right)}{\partial x}+a_7+\frac{\partial \left(c_2{a}_{c2}xy\right)}{\partial x}\right)+G{a}_9=0$$ (37)

$$2G\left(a_5+\frac{\partial \left(c_2{a}_{c2}xy\right)}{\partial y}\right)+\lambda \left(a_5+\frac{\partial \left(c_1{a}_{c1}xy\right)}{\partial y}+a_5+\frac{\partial \left(c_2{a}_{c2}xy\right)}{\partial y}\right)+G{a}_9=0$$ (38)

Integration of Eq. (37) with respect to x and Eq. (38) with respect to y gives:

$$2G\left(a_{7}x+c_1{a}_{c1}xy\right)+\lambda \left(c_1{a}_{c1}xy+2a_{7}x+c_2{a}_{c2}xy\right)+G{a}_{9}x+f\left(y\right)=0$$ (39)

$$2G\left(a_{5}y+c_2{a}_{c2}xy\right)+\lambda \left(2a_{5}y+c_1{a}_{c1}xy+c_2{a}_{c2}xy\right)+G{a}_{9}y+g\left(x\right)=0$$ (40)

Here, f(y) and g(x) are unknown functions that arise from the integrations. Solving Eqs. (39) and (40) simultaneously gives:

$$c_1=\left(\frac{\nu }{\left(1-\nu \right)x}\right)\left(\frac{a_5}{a_{c1}}\right)+\left(\frac{1}{\left(\nu -1\right)y}\right)\left(\frac{a_7}{a_{c1}}\right)+\left(\frac{\nu y-x}{2\left(1+\nu \right)xy}\right)\left(\frac{a_9}{a_{c1}}\right)+\frac{\nu g\left(x\right)-f\left(y\right)}{xyE}\left(\frac{1}{a_{c1}}\right)$$ (41)

$$c_2=\left(\frac{1}{\left(\nu -1\right)x}\right)\left(\frac{a_5}{a_{c2}}\right)+\left(\frac{\nu }{\left(1-\nu \right)y}\right)\left(\frac{a_7}{a_{c2}}\right)+\left(\frac{\nu x-y}{2\left(1+\nu \right)xy}\right)\left(\frac{a_9}{a_{c2}}\right)+\frac{\nu f\left(y\right)-g\left(x\right)}{xyE}\left(\frac{1}{a_{c2}}\right)$$ (42)

The unknown functions, f(y) and g(x), may take any form without violating compatibility and the force equilibrium within the element. However, the last terms in Eqs. (41) and (42), which contain the unknown functions, are not associated with any existing degrees of freedom, a_i, but temporary degrees of freedom ($a_{c1}$and $a_{c2}$), which will be eliminated. Therefore, the last terms must be equal to zero to ensure that all terms in the strain field are associated with an existing degree of freedom. Hence, $\nu g\left(x\right)-f\left(y\right)=0$ and $\nu f\left(y\right)-g\left(x\right)=0$, which gives $f\left(y\right)=0$ and $g\left(x\right)=0$. Eqs. (41) and (42) then simplify to:

$$c_1=\left(\frac{\nu }{\left(1-\nu \right)x}\right)\left(\frac{a_5}{a_{c1}}\right)+\left(\frac{1}{\left(\nu -1\right)y}\right)\left(\frac{a_7}{a_{c1}}\right)+\left(\frac{\nu y-x}{2\left(1+\nu \right)xy}\right)\left(\frac{a_9}{a_{c1}}\right)$$ (43)

$$c_2=\left(\frac{1}{\left(\nu -1\right)x}\right)\left(\frac{a_5}{a_{c2}}\right)+\left(\frac{\nu }{\left(1-\nu \right)y}\right)\left(\frac{a_7}{a_{c2}}\right)+\left(\frac{\nu x-y}{2\left(1+\nu \right)xy}\right)\left(\frac{a_9}{a_{c2}}\right)$$ (44)

Substituting Eqs. (43) and (44) into Eqs. (34) to (36) gives the final formulations that satisfy both compatibility and equilibrium conditions (temporary degrees of freedom a_c1, a_c2, and a_c3 will be eliminated upon the substitution):

$${\epsilon }_x=a_4+\left(\frac{y}{1-v}\right)a_5+\left(\frac{vx}{v-1}\right)a_7+\left(\frac{vy-x}{2\left(v+1\right)}\right)a_9$$ (45)

$${\epsilon }_y=\left(\frac{vy}{v-1}\right)a_5+a_6+\left(\frac{x}{1-v}\right)a_7+\left(\frac{vx-y}{2\left(v+1\right)}\right)a_9$$ (46)

$${\gamma }_{xy}=a_8+a_9\left(x+y\right)$$ (47)

An alternate or similar form can be obtained by using the second approach, that is, by considering all the corrective coefficients as constants. This approach avoids the need for the integration of the functions. Substituting Eqs. (34) to (36) into Eqs. (1) to (3) yields:

$$-c_3{a}_{c3}=0$$ (48)

$$2G\left(a_7+c_1{a}_{c1}y\right)+\lambda \left(a_7+c_1{a}_{c1}y+a_7+c_2{a}_{c2}y\right)+G\left(a_9+c_3{a}_{c3}x\right)=0$$ (49)

$$2G\left(a_5+c_2{a}_{c2}x\right)+\lambda \left(a_5+c_1{a}_{c1}x+a_5+c_2{a}_{c2}x\right)+G\left(a_9+c_3{a}_{c3}y\right)=0$$ (50)

Solving Eqs. (48) to (50) simultaneously gives the same expressions for the corrective coefficients that are shown in Eqs. (43) and (44), but with an additional solution for $c_3$.

Method validation

The first three formulations in Table 1 are implemented in the form of three-node triangular finite elements with three degrees of freedom for each node. Performances of TF₁, TF₂, and TF₃ are examined by solving three benchmark problems, which consist of domains with linear and non-linear sides. The problems considered are the deep cantilever beam, thick curved beam, and Cook's tapered beam, as shown in Fig. 1 (with selected examples of meshes used for the simulations). Meshes that are considered are provided in Table 2. These beams are subjected to uniform shear load at one end (the free end) while the other end is constrained. Displacements of point A for the benchmark problems are calculated and compared with the analytical solutions. Results are shown in Fig. 2.

Fig. 1.

Chosen benchmark problems (a) deep cantilever beam (E = $1\times {10}^{11}$, v = 0.2, thickness = 0.0625 and F = $1\times {10}^5$) (b) thick curved beam (E = 1000, v = 0, thickness = 1 and F = 600) (c) Cook's tapered beam (E = 1, v = 1/3, thickness = 1 and F = 1).

Table 2.

Mesh size and number of elements used for the simulations.

Thick curved beam		Deep cantilever beam		Cook's tapered beam
Mesh size	Number of elements	Mesh size	Number of elements	Mesh size	Number of elements
1 by 4	8	2 by 5	20	4 by 4	32
2 by 4	16	4 by 10	80	8 by 8	128
2 by 8	32	5 by 12	120	16 by 16	512
4 by 16	128	6 by 15	180	32 by 32	2048
5 by 32	320	8 by 20	320	–	–
8 by 32	512	–	–	–	–

Fig. 2.

Simulation results for the displacement of point A for the benchmark problems: (a) deep cantilever beam problem, (b) thick curved beam problem (c) Cook's tapered beam problem.

Results for TF₁(a) and TF₁(b) are reported as TF₁ (as general) in Fig. 2 since they yield the same results. Results for TF₂(b) are not available since it forms singular matrices due to the loss of degrees of freedom. Therefore, results for TF₂(a) are reported as TF₂ in Fig. 2. TF₂ contains higher-order terms (quadratic terms for the shear strain), while the rest of the formulations in Table 1 consist of first-order polynomial terms. Even though TF₃ contains first-order polynomial terms, it produces different results than TF₁. This finding indicates that accuracy and convergence of the solutions depend on the order of the polynomial terms that are present in the formulation and the sharing of the degrees of freedom among the strain components.

The corrective coefficients should be assigned accordingly to prevent the resultant strain functions from losing the independence of the degrees of freedom. For example, the independence of the degrees of freedom among the normal strain components for TF₃ and TF₄ is maintained in RF₃ and RF₄, respectively, by assigning the corrective coefficients to only the shear strain component. On the other hand, the independence of the degrees of freedom in the resultant strain functions will be lost if the corrective coefficients are assigned to two or all three strain components, as demonstrated by RF₁ and RF₂ (corresponding to TF₁ and TF₂, respectively).

RF₁(b) is found to have the same form as the one reported in a study by Belarbi et al. [30]; therefore, their performances are also the same. It is also seen that the performance of RF₂(a) is exactly similar to the formulation by Fortas et al. [28], even though these two formulations have different forms. Both formulations have the same polynomial order, and the degrees of freedom are distributed similarly among the strain components. It is also realized that RF₄ is exactly similar to the formulations in Refs. [35,36]. Upon comparison, it is seen that the degree of freedom $a_8$ is shared among the normal strain components in RF₃, which could be one of the causes for the inaccuracies in solving the thick curved beam and Cook's tapered beam problems, apart from being a shortage of one degree of freedom. The effect of independence of degrees of freedom on the simulation results is further evidenced by the performance of RF₃ as compared to the finite elements in Refs. [35,36] or RF₄.

Advantages of the technique presented in Section 3 are:

• 1.It can incorporate both compatibility and equilibrium conditions into existing formulations with incomplete polynomial terms.
• 2.Higher-order formulations (with more degrees of freedom) can be formulated by using lower-order formulations as the initial test functions.

Limitations of the technique are:

• 1.It requires initial test functions. These test functions can be the existing formulations or formulated based on the Pascal triangle.
• 2.Some resultant functions can lead to singular matrices, even though they satisfy both compatibility and equilibrium conditions.

All the formulations in this work were carried out independently; therefore, the actual methods used by other studies to develop the formulations are still unknown. The following section describes quadtree-based triangular mesh generation.

Quadtree-based triangular mesh generation

One of the drawbacks of formulating finite elements with lower-order polynomials is that sufficient accuracy for certain applications can only be achieved with fine mesh. The reason is that many elements are required to represent the non-linear behavior of the field variables within the problem domain. However, lower-order finite elements can be used for various applications since higher-order finite elements tend to be less accurate when they are used to represent lower-order variations of field variables. Therefore, quadtree meshes can be used to overcome the need for fine mesh generation throughout the problem domain.

Quadtree meshes can be generated by using square or triangular elements. Quadtree mesh enables fine mesh to be created in regions where variation of the filed variables is high (or in regions where high accuracy is needed). In contrast, a coarse mesh is used for the rest of the region. Both fine and coarse meshed regions are then merged by using transition elements [37]. Transition elements consist of a mid-node at one of the sides of the element. An additional shape function or formulation is needed to cater for the additional mid-node. Transition elements for strain-based finite elements are usually formed by allocating some degrees of freedom for the mid-node. Therefore, three-node strain-based triangular elements are classified as non-transitional elements [20].

This work introduces a new approach to form strain-based triangular transition elements (denoted as SB-TTE) by using three-node triangular elements. The fourth mid-node is formed by using virtual node method [37,38]; therefore, the need for more degrees of freedom is avoided. However, this approach requires formulation of the strain-based finite elements to be represented in the form of conventional displacement-based elements: by deriving shape functions. Strain-based finite element equations in terms of shape functions are provided in Section 2.1. Following section describes formulation of SB-TTE and application of the element in quadtree-based triangular mesh generation.

Strain-based triangular transition element (SB-TTE)

Virtual node method can be applied to form the four-node SB-TTE element. The transition element is formulated based on the subdivision of the parent element, which is a three-node triangular element with three degrees of freedom per node. The parent triangular element (with Nodes 1, 2, and 3) and subdivision of the element into four triangles (T₁, T₂, T₃, and T₄) are shown in Fig. 3.

Fig. 3.

A four-node SB-TTE element.

Dividing the parent element into four triangles leads to formation of two additional nodes, Node 4 and a virtual node, Node 5. Node 4 is in the middle of Side 1–3, while coordinates for the Node 5 (virtual node) are determined by using the formula below:

$$\left(x_5,y_5\right)=\frac{{\sum }_{i=1}^4\left(x_i,y_i\right)}{4}$$ (51)

The field variable matrix, $\left\{q\right\}$ for a particular triangular division or element with nodes i, j, and k ($T_{ijk}$) is given as:

$${\left\{q\right\}}^{\left[T_{ijk}\right]}={\left[N\right]}^{\left[T_{ijk}\right]}.{\left\{Q\right\}}^{\left[T_{ijk}\right]}$$ (52)

For example, the displacement $U$ for subdivision $T_{345}$ is given as:

$$\begin{array}{ccc}\hfill U^{\left[T_{345}\right]}& =& U_3{\left(N_1^U\right)}^{\left[T_{345}\right]}+U_4{\left(N_2^U\right)}^{\left[T_{345}\right]}+U_5{\left(N_3^U\right)}^{\left[T_{345}\right]}+V_3{\left(N_4^U\right)}^{\left[T_{345}\right]}+V_4{\left(N_5^U\right)}^{\left[T_{345}\right]}+V_5{\left(N_6^U\right)}^{\left[T_{345}\right]}\hfill \\ & & +{\theta }_3{\left(N_7^U\right)}^{\left[T_{345}\right]}+{\theta }_4{\left(N_8^U\right)}^{\left[T_{345}\right]}+{\theta }_5{\left(N_9^U\right)}^{\left[T_{345}\right]}\hfill \end{array}$$ (53)

Field variables at the virtual node are given as:

$${\left\{U_5,V_5,{\theta }_5\right\}}^T={\left[N\left(x_5,y_5\right)\right]}^{\left[T_{123}\right]}{\left\{Q\right\}}^{\left[T_{123}\right]}$$ (54)

Substitution of Eq. (54) above into Eq. (53) eliminates the field variables at the virtual nodes from Eq. (53). Next, the SB-TTE elements are used in the first benchmark problem (deep cantilever beam) to generate two quadtree-based triangular meshes (Mesh 1 and Mesh 2), as shown in Fig. 4. Mesh 1 consists of 44 elements with 34 global nodes, while Mesh 2 consists of 200 elements with 124 global nodes. The problem is solved by using formulation proposed in Ref. [11] and compared with the solutions provided for uniform meshes in Ref. [11]. Analytical solution is 1.105 mm and the results are provided in Table 3. Simulation result in Fig. 5 shows that SB-TTE can be used for quadtree-based triangular mesh generation. Compared to uniform meshes, faster convergence to the analytical solution is seen for quadtree-based triangular meshes, producing more accurate results.

Fig. 4.

Quadtree-based triangular meshes with SB-TTE (a) Mesh 1 (b) Mesh 2.

Table 3.

Mesh size and number of elements used for the simulations.

Quadtree-based Mesh		Uniform mesh
Mesh	Displacement (in mm)	Mesh	Displacement (in mm)
Mesh 1	1.054	6 by 15	1.056
Mesh 2	1.107	8 by 20	1.086

Fig. 5.

Simulation results for quadtree-based triangular meshes and uniform meshes.

Conclusion

New techniques for element formulation and quadtree-based triangular mesh generation have been successfully developed and tested for strain-based finite elements. This work has shown that use of corrective coefficients ensures that the final formulations satisfy both compatibility and equilibrium conditions. This novel technique helps to avoid vague trial and error procedures that are used in some of the original methods (provided in the specification table) for the element formulation. Advantages and limitations of the method are provided in section 3.1. The technique can be easily implemented for three-dimensional cases as well. Two different approaches are shown for the calculation of the stiffness matrix. The first approach (conventional approach for strain-based elements) involves the inversion of matrices shown in Eq. (26). In contrast, the second (alternate) approach involves the derivation of shape functions Eqs. (32) and ((33)). Even though both approaches yield the same results, the alternate approach is useful for quadtree-based triangular mesh generation since nodal shape functions are needed for the particular meshing technique. Besides, quadtree-based triangular meshes exhibit more rapid convergence compared to uniform meshes. It is also shown that three-node strain-based triangular elements can be used as transition elements by using the virtual node method without adding more degrees of freedom. Strain-based elements that are formulated based on the techniques described in this paper can be used to solve solid mechanics problems. The technique has potential to be implemented in other engineering fields as well, if proper conditions to ensure accuracy and convergence of the results for the selected application can be determined. Current work includes formulating polygonal elements by using the techniques described in this paper.

CRediT authorship contribution statement

Logah Perumal: Conceptualization, Methodology, Software, Formal analysis, Investigation, Resources, Writing – original draft, Supervision, Funding acquisition. Wei Hao Koh: Software, Validation, Formal analysis, Resources, Writing – review & editing, Visualization.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Data availability

• No data was used for the research described in the article.