Standard Linear Element Research Articles

Linear viscoelastic response of the vertex model with internal and external dissipation: Normal modes analysis

We use the normal mode formalism to study the shear rheology of the vertex model for epithelial tissue mechanics in the overdamped linear response regime. We consider systems with external (e.g., cell-substrate) and internal (e.g., cell-cell) dissipation mechanisms, and derive expressions for stresses on cells due to mechanical and dissipative forces. The semi-analytical method developed here is, however, general and can be directly applied to study the linear response of a broad class of soft matter systems with internal and external dissipation. It involves normal mode decomposition to calculate linear loss and storage moduli of the system. Specifically, displacements along each normal mode produce stresses due to elastic deformation and internal dissipation, which are in force balance with loads due to external dissipation. Each normal mode responds with a characteristic relaxation timescale, and its rheological behavior can be described as a combination of a standard linear solid element due to elastic stresses and a Jeffreys model element due to the internal dissipative stresses. The total response of the system is then fully determined by connecting in parallel all the viscoelastic elements corresponding to individual normal modes. This allows full characterization of the potentially complex linear rheological response of the system at all driving frequencies and identification of collective excitations. We show that internal and external dissipation mechanisms lead to qualitatively different rheological behaviors due to the presence or absence of Jeffreys elements, which is particularly pronounced at high driving frequencies. Our findings, therefore, underscore the importance of microscopic dissipation mechanisms in understanding the rheological behavior of soft materials and tissues, in particular.

On the improvement of the triangular Shepard method by non conformal polynomial elements

In this paper, we introduce a new nonconforming finite element as a polynomial enrichment of the standard triangular linear element. Based on this new element, we propose an improvement of the triangular Shepard operator. We prove that the order of this new approximation operator is at least cubic. Numerical experiments demonstrate the accuracy of the proposed method.

An accurate, robust, and efficient finite element framework with applications to anisotropic, nearly and fully incompressible elasticity

Fiber-reinforced soft biological tissues are typically modeled as hyperelastic, anisotropic, and nearly incompressible materials. To enforce incompressibility a multiplicative split of the deformation gradient into a volumetric and an isochoric part is a very common approach. However, the finite element analysis of such problems often suffers from severe volumetric locking effects and numerical instabilities. In this paper, we present novel methods to overcome volumetric locking phenomena for using stabilized P1–P1 elements. We introduce different stabilization techniques and demonstrate the high robustness and computational efficiency of the chosen methods. In two benchmark problems from the literature as well as an advanced application to cardiac electromechanics, we compare the approach to standard linear elements and show the accuracy and versatility of the methods to simulate anisotropic, nearly and fully incompressible materials. We demonstrate the potential of this numerical framework to accelerate accurate simulations of biological tissues to the extent of enabling patient-specific parameterization studies, where numerous forward simulations are required.

Implicit and semi-implicit second-order time stepping methods for the Richards equation

This study concerns numerical methods for efficiently solving the Richards equation where different weak formulations and computational techniques are analyzed. The spatial discretizations are based on standard or mixed finite element methods. Different implicit and semi-implicit temporal discretization techniques of second-order accuracy are studied. To obtain a linear system for the semi-implicit schemes, we propose second-order techniques using extrapolation formulas and/or semi-implicit Taylor approximations for the temporal discretization of nonlinear terms. A numerical convergence study and a series of numerical tests are performed to analyze efficiency and robustness of the different schemes. The developed scheme, based on the proposed temporal extrapolation techniques and the mixed formulation involving the saturation and pressure head and using the standard linear Lagrange element, performs better than other schemes based on the saturation and the flux and using the Raviart-Thomas elements. The proposed semi-implicit scheme is a good alternative when implicit schemes meet convergence issues.

Three Numerical Eigensolvers for 3-D Cavity Resonators Filled With Anisotropic and Nonconductive Media

This paper mainly investigates the classic resonant cavity problem with anisotropic and nonconductive media, which is a linear vector Maxwell's eigenvalue problem. The finite element method based on edge element of the lowest-order and standard linear element is used to solve this type of 3-D closed cavity problem. In order to eliminate spurious zero modes in the numerical simulation, the divergence-free condition supported by Gauss' law is enforced in a weak sense. After the finite element discretization, the generalized eigenvalue problem with a linear constraint condition needs to be solved. Penalty method, augmented method and projection method are applied to solve this difficult problem in numerical linear algebra. The advantages and disadvantages of these three computational methods are also given in this paper. Furthermore, we prove that the augmented method is free of spurious modes as long as the anisotropic material is not magnetic lossy. The projection method based on singular value decomposition technique can be used to solve the resonant cavity problem. Moreover, the projection method {cannot} introduce any spurious modes. At last, several numerical experiments are carried out to verify our theoretical results.

A counterexample for the inf‐sup stability of the RT0 − P1 ⊂ L2(Ω) × H10(Ω) finite element combination for the mixed Poisson equation

AbstractThis paper investigates the inf‐sup stability of a dual mixed discretization of the Poisson problem using the lowest‐order (Hdiv‐conforming) Raviart–Thomas space combined with the space of standard linear (continuous) Lagrange elements.

A Nitsche-eXtended Finite Element Method for Distributed Optimal Control Problems of Elliptic Interface Equations

Abstract This paper analyzes an interface-unfitted numerical method for distributed optimal control problems governed by elliptic interface equations. We follow the variational discretization concept to discretize the optimal control problems and apply a Nitsche-eXtended finite element method to discretize the corresponding state and adjoint equations, where piecewise cut basis functions around the interface are enriched into the standard linear element space. Optimal error estimates of the state, co-state and control in a mesh-dependent norm and the L 2 {L^{2}} norm are derived. Numerical results are provided to verify the theoretical results.

Novel quadratic Bézier triangular and tetrahedral elements using existing mesh generators: Applications to linear nearly incompressible elastostatics and implicit and explicit elastodynamics

SummaryIn this paper, we present novel techniques of using quadratic Bézier triangular and tetrahedral elements for elastostatic and implicit/explicit elastodynamic simulations involving nearly incompressible linear elastic materials. A simple linear mapping is proposed for developing finite element meshes with quadratic Bézier triangular/tetrahedral elements from the corresponding quadratic Lagrange elements that can be easily generated using the existing mesh generators. Numerical issues arising in the case of nearly incompressible materials are addressed using the consistent B‐bar formulation, thus reducing the finite element formulation to one consisting only of displacements. The higher‐order spatial discretization and the nonnegative nature of Bernstein polynomials are shown to yield significant computational benefits. The optimal spatial convergence of the B‐bar formulation for the quadratic triangular and tetrahedral elements is demonstrated by computing error norms in displacement and stresses. The applicability and computational efficiency of the proposed elements for elastodynamic simulations are demonstrated by studying several numerical examples involving real‐world geometries with complex features. Numerical results obtained with the standard linear triangular and tetrahedral elements are also presented for comparison.

Displacement and Stress Function-based Linear and Quadratic Triangular Elements for Saint-Venant Torsional Problems

Torsional problems commonly arise in frame structural members subjected to unsym­metrical loading. Saint-Venant proposed a semi inverse method to develop the exact theory of torsional bars of general cross sections. However, the solution to the problem using an analytical method for a complicated cross section is cumbersome. This paper presents the adoption of the Saint-Venant theory to develop a simple finite element program based on the displacement and stress function approaches using the standard linear and quadratic triangular elements. The displacement based approach is capable of evaluating torsional rigidity and shear stress distribution of homogeneous and nonhomogeneous; isotropic, orthotropic, and anisotropic materials; in singly and multiply-connected sections. On the other hand, applications of the stress function approach are limited to the case of singly-connected isotropic sections only, due to the complexity on the boundary conditions. The results show that both approaches converge to exact solutions with high degree of accuracy.

Scaled boundary finite element method with exact defining curves for two-dimensional linear multi-field media

This paper presents an efficient and accurate numerical technique based upon the scaled boundary finite element method for the analysis of two-dimensional, linear, second-order, boundary value problems with a domain completely described by a circular defining curve. The scaled boundary finite element formulation is established in a general framework allowing single-field and multi-field problems, bounded and unbounded bodies, distributed body source, and general boundary conditions to be treated in a unified fashion. The conventional polar coordinates together with a properly selected scaling center are utilized to achieve the exact description of the circular defining curve, exact geometry of the domain, and exact spatial differential operators. Standard finite element shape functions are employed in the discretization of both trial and test functions in the circumferential direction and the resulting eigenproblem is solved by a selected efficient algorithm. The computational performance of the implemented procedure is then fully investigated for various scenarios to demonstrate the accuracy in comparison with standard linear elements.

Numerical investigation of circle defining curve for two-dimensional problem with general boundaries using the scaled boundary finite element method

The scaled boundary finite element method (SBFEM) is applied to the static analysis of two dimensional elasticity problem, boundary value problems domain with the domain completely described by a circular defining curve. The scaled boundary finite element equations is formulated within a general framework integrating the influence of the distributed body force, general boundary conditions, and bounded and unbounded domain. This paper investigates the possibility of using exact geometry to form the exact description of the circular defining curve and the standard finite element shape function to approximate the defining curve. Three linear elasticity problems are presented to verify the proposed method with the analytical solution. Numerical examples show the accuracy and efficiency of the proposed method, and the performance is found to be better than using standard linear element for the approximation defining curve on the scaled boundary method.

Mixed Finite-Element Method for the Closed Waveguide Problem Filled With Anisotropic Media

A mixed finite-element method based on constant transversal and linear normal element and standard linear element, linear transversal and quadratic normal element, and standard quadratic element is used to solve the closed waveguide problem filled with anisotropic media. This numerical method is free of nonphysical modes for solving the closed waveguide problem, which is governed by both electric field and magnetic field. Numerical experiments show that the numerical propagation constants for TEM modes are of high accuracy, and the numerical method given in this paper can support an effective interval for the exact propagation constants of hybrid modes in the waveguide; hence, it is easy to know the numerical accuracy of the approximate propagation constants for the hybrid modes.

Finite Element Method for Resonant Cavity Problem With Complex Geometrical Structure and Anisotropic Fully Conducting Media

In this paper, the resonant cavity problem with anisotropic fully conducting media, complex geometrical structure and perfect electric conductor walls is investigated. We solve this problem based on the finite element method (FEM) with tangential and linear normal (CT/LN) element and standard linear element. An effective numerical method is proposed by us such that it is free of nonphysical modes. After the FEM discretization, we need to solve a quadratic algebraic eigenvalue problem with a linear constraint condition. In order to overcome this difficulty in the field of numerical algebra, we change this algebraic eigenvalue problem into a generalized eigenvalue problem by introducing an auxiliary zero eigenvector. Moreover, when the permittivity and conductivity are two constants, both the eigenmodes of infinite algebraic multiplicity and all the nonphysical modes are also removed by linearization method. Several numerical experiments show that computational method in this paper can suppress all the spurious modes.

Robust and efficient validation of the linear hexahedral element

Checking mesh validity is a mandatory step before doing any finite element analysis. If checking the validity of tetrahedra is trivial, checking the validity of hexahedral elements is far from being obvious. In this paper, a method that robustly and efficiently compute the validity of standard linear hexahedral elements is presented. This method is a significant improvement of a previous work on the validity of curvilinear elements [1]. The new implementation is simple and computationally efficient. The key of the algorithm is still to compute Bézier coefficients of the Jacobian determinant. We show that only 20 Jacobian determinants are necessary to compute the 27 Bézier coefficients. Those 20 Jacobians can be efficiently computed by calculating the volume of 20 tetrahedra. The new implementation is able to check the validity of about 6 million hexahedra per second on one core of a personal computer. Through the paper, all the necessary information is provided that allow to easily reproduce the results, i.e. write a simple code that takes the coordinates of 8 points as input and outputs the validity of the hexahedron.

Investigation of size-dependent quasistatic response of electrically actuated nonlinear viscoelastic microcantilevers and microbridges

This study investigates the size-dependent quasistatic response of a nonlinear viscoelastic microelectromechanical system (MEMS) under an electric actuation. To have this problem in view, the deformable electrode of the MEMS is modelled using cantilever and doubly-clamped viscoelastic microbeams. The modified couple stress theory in conjunction with Bernoulli–Euler beam theory are used for mathematical modeling of the size-dependent instability of microsystems in the framework of linear viscoelastic theory. Simultaneous effect of electrostatic actuation including fringing field, residual stress, mid-plane stretching and Casimir and van der Waals intermolecular forces are considered in the theoretical model. A single element of the standard linear solid element is used to simulate the viscoelastic behavior. Based on the extended Hamilton’s variational principle, the nonlinear governing integro-differential equation and boundary conditions are derived. Thereafter, a new generalized differential-integral quadrature solution for the nonlinear quasistatic response of electrically actuated viscoelastic micro/nanobeams under two different boundary conditions; doubly-clamped microbridge and clamped-free microcantilever. The developed model is verified and a good agreement is obtained. Finally, a comprehensive study is conducted to investigate the effects of various parameters such as material relaxation time, durable modulus, material length scale parameter, Casimir force, van der Waals force, initial gap and beam length on the pull-in response of viscoelastic microbridges and microcantilevers in the framework of viscoelasticity.

Adaptive flux-based nodeless variable finite element formulation with error estimation for thermal-structural analysis

A posteriori error estimation for the nodeless variable finite element method is presented. A nodeless variable finite element method using flux-based formulation is developed and combined with an adaptive meshing technique to analyze two-dimensional thermal-structural problems. The continuous flux and stresses are determined by using the flux-based formulation while the standard linear element interpolation functions are used to determine the discontinuous flux and stresses. To measure the global error, the L2 norm error is selected to find the root-mean-square error over the entire domain. The finite element formulation and its detailed finite element matrices are presented. Accuracy of the estimated error is measured by the percentage relative error. An adaptive meshing technique, that can generate meshes corresponding to solution behaviors automatically, is implemented to further improve the solution accuracy. Several examples are presented to evaluate the performance and accuracy of the combined method.

Three-dimensional isogeometrically enriched finite elements for frictional contact and mixed-mode debonding

We present an isogeometric enrichment technique for three-dimensional finite element computations applied to frictional contact and mixed-mode debonding. This is an extension of previous work that focused on two-dimensional and frictionless problems. To offer a more complete view of the enriched element’s performance, a comparison of the results to tri-variate isogeometric discretizations and standard Lagrangian elements is also included here. The enrichment is applied by discretizing parts of the surface that require higher accuracy with isogeometric basis functions, while the rest of the body uses Lagrangian shape functions. By using an isogeometric surface representation, the higher continuity across element boundaries and higher order of interpolation can be exploited. At the same time, the generation of tri-variate isogeometric meshes is avoided.A convergence study without any surface effects, involving only volume integrals, shows that the enriched elements can also be beneficial for these problems. The major advantage of the isogeometric element enrichment over standard tri-linear elements is demonstrated in contact problems including normal and tangential tractions. For both, mixed-mode cohesive debonding and frictional contact, the enrichment increases robustness and leads to more accurate results than standard linear Lagrangian elements. All computations are also compared to results using tri-variate isogeometric discretizations to give a complete picture of the element’s performance. It is also shown that the proposed enrichment formulation has advantages in mesh generation.

Accurate finite element modeling of acoustic waves

In the paper we suggest an accurate finite element approach for the modeling of acoustic waves under a suddenly applied load. We consider the standard linear elements and the linear elements with reduced dispersion for the space discretization as well as the explicit central-difference method for time integration. The analytical study of the numerical dispersion shows that the most accurate results can be obtained with the time increments close to the stability limit. However, even in this case and the use of the linear elements with reduced dispersion, mesh refinement leads to divergent numerical results for acoustic waves under a suddenly applied load. This is explained by large spurious high-frequency oscillations. For the quantification and the suppression of spurious oscillations, we have modified and applied a two-stage time-integration technique that includes the stage of basic computations and the filtering stage. This technique allows accurate convergent results at mesh refinement as well as significantly reduces the numerical anisotropy of solutions. We should mention that the approach suggested is very general and can be equally applied to any loading as well as for any space-discretization technique and any explicit or implicit time-integration method.

The Study on 3D Incompatible Element in Flexible Multibody System Dynamics

In this study the performance of three-dimensional tetrahedron elements has been compared with the background of the simulation of Flexible Multibody System (FMBS). The standard linear tetrahedron element has the least number of nodes, however it has the locking problem. The standard quadratic element has shown precise result and robust performance in many applications, but its high nodes also means large computational time. The incompatible elements integrates the excellence of the linear and quadratic elements and its excellent performance in term of accuracy and time efficiency is presented.

Finite element modeling of linear elastodynamics problems with explicit time-integration methods and linear elements with the reduced dispersion error

We have developed two finite element techniques with reduced dispersion for linear elastodynamics that are used with explicit time-integration methods. These techniques are based on the modified integration rule for the mass and stiffness matrices and on the averaged mass matrix approaches that lead to the numerical dispersion reduction for linear finite elements. The analytical study of numerical dispersion for the new techniques is carried out in the 1-D, 2-D and 3-D cases. The numerical study of the efficiency of the dispersion reduction techniques includes the two-stage time-integration approach with the filtering stage (developed in our previous papers) that quantifies and removes spurious high-frequency oscillations from numerical results. We have found that in contrast to the standard linear elements with explicit time-integration methods and the lumped mass matrix, the finite element techniques with reduced dispersion yield more accurate results at small time increments (smaller than the stability limit) in the 2-D and 3-D cases. The recommendations for the selection of the size of time increments are suggested. The new approaches with reduced dispersion can be easily implemented into existing finite element codes and lead to significant reduction in computation time at the same accuracy compared with the standard finite element formulations.