Incompressible Elasticity Problems Research Articles

An FFT-Based MAC scheme for Stokes equations with periodic boundary conditions and its application to elasticity problems

This paper presents a fast Fourier transform (FFT) based finite difference MAC scheme to compute the Stokes equations with periodic boundary conditions. The numerical implementation of this solver is very convenient and only requires three simple steps. The remarkable superiorities of such a solver compared to classical iterative approaches are computational time-saving and memory storage saving. The application of the proposed solver to nearly incompressible linear elasticity problems is also introduced. In addition, following the similar spirit used in Dong et al. (2020) for Stokes equations with Dirichlet boundary conditions, a rigorous error analysis is provided for Stokes equations with periodic boundary conditions. The results include: a second-order convergence in the ℓ2-norm for velocity, pressure as well as the gradient of velocity; a second-order accuracy in maximum norm for both velocity and its gradient. Moreover, the properties of the discrete Green functions play an important part in maximum error estimation. Most notably, the major difficulties of the extension from Dirichlet boundary conditions to periodic boundary conditions are the discrete version of Green’s formulae, Poincaré inequality and LBB condition. A variety of two-dimensional and three-dimensional numerical examples are given to illustrate the efficiency of the proposed FFT solver and validate the theoretical results.

A subspace of linear nonconforming finite element for nearly incompressible elasticity and Stokes flow

Abstract The linear nonconforming finite element, combined with constant finite element for pressure, is stable for the Stokes problem. But it does not satisfy the discrete Korn inequality. The linear conforming finite element satisfies the discrete Korn inequality, but is not stable for the Stokes problem and fails for the nearly incompressible elasticity problems. We enrich the linear conforming finite element by some nonconforming P1 bubbles, i.e., select a subspace of the linear nonconforming finite element space, so that the resulting linear nonconforming element is both stable and conforming enough to satisfy the Korn inequality, on HTC-type triangular and tetrahedral grids. Numerical tests in 2D and 3D are presented, confirming the analysis.

Numerical analysis of a stabilized scheme applied to incompressible elasticity problems with Dirichlet and with mixed boundary conditions

Numerical analysis of a stabilized scheme applied to incompressible elasticity problems with Dirichlet and with mixed boundary conditions

General finite-volume framework for saddle-point problems of various physics

AbstractThis article is dedicated to the general finite-volume framework used to discretize and solve saddle-point problems of various physics. The framework applies the Ostrogradsky–Gauss theorem to transform a divergent part of the partial differential equation into a surface integral, approximated by the summation of vector fluxes over interfaces. The interface vector fluxes are reconstructed using the harmonic averaging point concept resulting in the unique vector flux even in a heterogeneous anisotropic medium. The vector flux is modified with the consideration of eigenvalues in matrix coefficients at vector unknowns to address both the hyperbolic and saddle-point problems, causing nonphysical oscillations and an inf-sup stability issue. We apply the framework to several problems of various physics, namely incompressible elasticity problem, incompressible Navier–Stokes, Brinkman–Hazen–Dupuit–Darcy, Biot, and Maxwell equations and explain several nuances of the application. Finally, we test the framework on simple analytical solutions.

A nonlinear plane strain finite element analysis for multilayer elastomeric bearings

The purpose of this paper is the prediction of the behaviour of multilayer elastomeric bearings by the finite element analysis. First, the finite element formulation of equilibrium equations of incompressible elasticity problem is presented. This approach is based on the modified potential energy taking into account the incompressibility. The linearized system obtained is solved by the Newton–Raphson iterative method with the total lagrangian formulation. A large plane strain element is developed for the analysis of elastomeric materials under large strain deformation. This element is then integrated into our software. After validating the plane strain element using examples with known exact solutions, an industrial case study made of laminated (metal-rubber) elastomeric component is examined. The stress distribution study is limited to the elastomeric layers, possible sources of crack appearance.

Preconditioned Conjugate Gradient Methods for the Refined FEM Discretizations of Nearly Incompressible Elasticity Problems in Three Dimensions

Nearly incompressible problems in three dimensions are the important problems in practical engineering computation. The volume-locking phenomenon will appear when the commonly used finite elements such as linear elements are applied to the solution of these problems. There are many efficient approaches to overcome this locking phenomenon, one of which is the higher-order conforming finite element method. However, we often use the lower-order nonconforming elements as Wilson elements by considering the computational complexity for three-dimensional (3D) problems considered. In general, the convergence of Wilson elements will heavily rely on the quality of the meshes. It will greatly deteriorate or no longer converge when the mesh distortion is very large. In this paper, the refined element method based on Wilson element is first applied to solve nearly incompressible elasticity problems, and the influence of mesh quality on the refined element is tested numerically. Its validity is verified by some numerical examples. By using the internal condensation method, the refined element discrete system of equations is deduced into the one which is spectrally equivalent to an 8-node hexahedral element discrete system of equations. And then, a type of efficient algebraic multigrid (AMG) preconditioner is presented by combining both the coarsening techniques based on the distance matrix and the effective smoothing operators. The resulting preconditioned conjugate gradient (PCG) method is efficient for 3D nearly incompressible problems. The numerical results verify the efficiency and robustness of the proposed method.

A Locking-Free DP-Q2-P1 MFEM for Incompressible Nonlinear Elasticity Problems

A Locking-Free DP-Q2-P1 MFEM for Incompressible Nonlinear Elasticity Problems

Robust Preconditioning for Stochastic Galerkin Formulations of Parameter-Dependent Nearly Incompressible Elasticity Equations

We consider the nearly incompressible linear elasticity problem with an uncertain spatially varying Young's modulus. The uncertainty is modeled with a finite set of parameters with prescribed proba...

A Stabilized Hybrid Discontinuous Galerkin Method for Nearly Incompressible Linear Elasticity Problem

In this work, a primal hybrid finite element method for nearly incom pressible linear elasticity problem on triangular meshes is shown. This method consists of coupling local discontinuous Galerkin problems to the primal variable with a global problem for the Lagrange multiplier, which is identified as the trace of the displacement field. Also, a local post-processing technique is used to recover stress approximations with improved rates of convergence in H(div) norm. Numerical studies show that the method is locking free even using equal or different orders for displacement and stress fields and optimal convergence rates are obtained.

Numerical assessment of two-level domain decomposition preconditioners for incompressible Stokes and elasticity equations

Solving the linear elasticity and Stokes equations by an optimal domain decomposition method derived algebraically involves the use of non standard interface conditions. The one-level domain decomposition preconditioners are based on the solution of local problems. This has the undesired consequence that the results are not scalable, it means that the number of iterations needed to reach convergence increases with the number of subdomains. This is the reason why in this work we introduce, and test numerically, two-level preconditioners. Such preconditioners use a coarse space in their construction. We consider the nearly incompressible elasticity problems and Stokes equations, and discretise them by using two finite element methods, namely, the hybrid discontinuous Galerkin and Taylor-Hood discretisations.

A Stabilized Hybrid Discontinuous Galerkin Method for Nearly Incompressible Linear Elasticity Problem

A Stabilized Hybrid Discontinuous Galerkin Method for Nearly Incompressible Linear Elasticity Problem

A low-order finite element method for three dimensional linear elasticity problems with general meshes

The paper is concerned with a low-order finite element method, namely the staggered cell-centered finite element method, which has been proposed and analyzed in Ong et al. (2015) for two-dimensional compressible and nearly incompressible linear elasticity problems. In this work, we extend the results to the three-dimensional case and focus on the creating of the meshes. In particular, from a general primal mesh M, we construct a polygonal dual mesh M∗ and its submesh M∗∗ in a way such that each dual control volume of M∗ corresponds to a primal vertex and is a union (macro-element) of some fixed number of adjacent tetrahedral elements of M∗∗. The displacement is approximated by piecewise trilinear functions on the subdual mesh M∗∗ and the pressure by piecewise constant functions on the dual mesh M∗. As for two-dimensional case, such construction of the meshes and approximation spaces satisfies the macroelement condition, which implies stability and convergence of the scheme. Numerical experiments are carried out to investigate the performance of the proposed method on various mesh types.

A conservative stable finite element method for Stokes flow and nearly incompressible linear elasticity on rectangular grid

In this paper, we discuss finite element methods for the incompressible Stokes problem and the nearly incompressible linear elasticity problem. Specifically, we present a finite element pair for the incompressible Stokes problem, which satisfies the discrete inf–sup condition and the discrete Korn’s inequality, and moreover, which is element-wise conservative. The pair provides a locking-free method for the nearly incompressible linear elasticity problem without reduced integration.

Preconditioning Methods for the Penalty Function FEM Discretizations of 3D Nearly Incompressible Elasticity Problems

Penalty function finite element method (FEM) is an efficient method for the solution of the nearly incompressible elasticity problems in three dimensions. In order to improve the overall efficiency, some faster solvers for the corresponding FEM system of equations must be presented. Since the resulting system of equations is symmetric positive definite and highly ill-conditioned, the preconditioned conjugate gradient (PCG) method is one of the most efficient methods for solving such FEM equations. In this paper, we have first presented some types of efficient PCG methods including [Formula: see text]-PCG and [Formula: see text]-PCG with two different block diagonal inverses as a preconditioner, respectively, and including the RS-GAMG-PCG based on the global matrix for the penalty function FEM equations of three-dimensional (3D) nearly incompressible elasticity problems with mixed boundary conditions. Furthermore, we apply these methods to the solution of the penalty function quadratic discretizations for the cantilever and the Cook’s membrane problems. The numerical results have shown that much more efficient PCG methods for 3D nearly incompressible problems can be obtained by using the known information that is easily available in most FEM applications, for instance, the type of the partial differential equations (PDEs) considered and the number of physical unknowns residing in each grid, and by combining the coarsening techniques used in the classic algebraic multigrid (AMG) method. This will greatly improve the overall efficiency of the penalty function FEM analysis.

A Staggered Cell-Centered Finite Element Method for Compressible and Nearly-Incompressible Linear Elasticity on General Meshes

We propose a new numerical method, namely, the staggered cell-centered finite element method for compressible and nearly incompressible linear elasticity problems. By building a dual mesh and its triangular submesh, the scheme can be constructed from a general mesh in which the displacement is approximated by piecewise linear (P1) functions on the dual submesh and, in the case of nearly incompressible problems, the pressure is approximated by piecewise constant (P0) functions on the dual mesh. The scheme is cell centered in the sense that the solution can be computed by cell unknowns of the primal mesh (for the displacement) and of the dual mesh (for the pressure). The method is presented within a rigorous theoretical framework to show stability and convergence. In particular, for the nearly incompressible case, stability is proved by using the macroelement technique. Numerical results show that the method, compared with other methods, is effective in terms of accuracy and computational cost.

Parallel subspace correction methods for nearly singular systems

In this paper we consider the parallel subspace correction (PSC) methods for the nearly singular systems. We apply the PSC methods as the preconditioners when we solve the nearly singular systems by the conjugate gradient methods. Our focus is to estimate the condition number of the preconditioned systems. We deduce the parameter independent estimates on the PSC preconditioners for the nearly singular systems, under appropriate assumptions on subspace decomposition. The main assumption is that the kernel of the singular part of the system can be decomposed into a sum of local kernel subspaces. Our estimates can be applied into actual problems, and two examples are analyzed in this paper. One is the elliptic problem with large jumps in the coefficients, the other is the planar nearly incompressible elasticity problem with the Scott–Vogelius finite element discretization. We prove that the related parallel multilevel methods for both examples are convergent uniformly, with respect to the coefficients and the mesh size.

三维近不可压缩弹性问题的罚函数有限元分析

对三维近不可压缩弹性问题，利用常规有限元进行求解时会出现体积闭锁现象，需要采用某些特殊的方法。罚函数协调有限元法具有程序实现简单、罚数易于确定以及不改变泛函驻值性质等特点，是克服体积闭锁现象的一种有效方法。本文，针对混合边界条件的三维近不可压缩问题，详细推导了罚函数有限元法的计算格式，分析该方法实施成功的条件，并通过数值实验验证了该方法对解决体积闭锁现象的有效性和鲁棒性。在三维有限元分析中，剖分网格的质量将对计算精度和求解效率产生很大影响，实际计算时若能采用各向同性网格，则对问题的分析将具有更好的收敛性。The locking phenomenon will appear when the commonly used finite elements are applied to the solution of nearly incompressible problems in three dimensions. It is necessary to use some special methods. The penalty function conforming finite element method is an effective method to overcome this locking phenomenon since it is simple for the realization of the resulting program and easy to determine the penalty number and it also does not change the functional stationary value properties. In this paper, the computing format of penalty function finite element method is carefully derived, the conditions for success of the resulting method is analyzed and the effective-ness and robustness of this method are finally verified by some numerical experiments for nearly incompressible elasticity problems. The quality of the mesh used in three-dimensional finite ele-ment analysis has a great effect on the accuracy and computational efficiency. If the isotropic grids can be used in the practical calculations, the method will have better convergence.

Radial Basis Collocation Method for Nearly Incompressible Elasticity

AbstractStrong form collocation using radial basis functions approximation, which is called the radial basis collocation method, is an efficient method for solving partial differential equations or integral equations because it has implementation simplicity and exponential convergence. In this work, the radial basis collocation method is introduced in conjunction with a least-squares scheme to suppress the volumetric locking in nearly incompressible elastic problems. Radial basis approximation, which possesses infinite continuity, is rich enough to ensure the divergence-free condition. Modal analysis proves that all the nonphysical locking modes can be removed, which eliminates the locking effect. Numerical studies validate the good performance of the proposed method. Exponential convergence is obtained as expected, and no pressure oscillation is observed.

Deflation, Projector Preconditioning, and Balancing in Iterative Substructuring Methods: Connections and New Results

In this paper, projector preconditioning, also known as the deflation method, as well as the balancing preconditioner are applied to the dual-primal finite element tearing and interconnecting (FETI-DP) and balancing domain decomposition by constraints (BDDC) methods in order to create a second, independent coarse problem. This may help to extend the parallel scalability of classical FETI-DP and BDDC methods without the use of inexact solvers and may also be used to improve the robustness, e.g., for almost incompressible elasticity problems. Connections of FETI-DP methods applying a transformation of basis using a larger coarse space with a corresponding FETI-DP method using projector preconditioning or balancing are pointed out. It is then shown that the methods have essentially the same spectrum. Numerical results for compressible and almost incompressible linear elasticity are provided. The sensitivity of the projection methods to an inexact computation of the projections is numerically investigated and a different behavior for projector preconditioning and the balancing preconditioner is found.

A Note on the Effect of the Choice of Weak Form on GMRES Convergence for Incompressible Nonlinear Elasticity Problems

The generalized minimal residual (GMRES) method is a common choice for solving the large nonsymmetric linear systems that arise when numerically computing solutions of incompressible nonlinear elasticity problems using the finite element method. Analytic results on the performance of GMRES are available on linear problems such as linear elasticity or Stokes’ flow (where the matrices in the corresponding linear systems are symmetric), or on the nonlinear problem of the Navier–Stokes flow (where the matrix is block-symmetric/block-skew-symmetric); however, there has been very little investigation into the GMRES performance in incompressible nonlinear elasticity problems, where the nonlinearity of the incompressibility constraint means the matrix is not block-symmetric/block-skew-symmetric. In this short paper, we identify one feature of the problem formulation, which has a huge impact on unpreconditioned GMRES convergence. We explain that it is important to ensure that the matrices are perturbations of a block-skew-symmetric matrix rather than a perturbation of a block-symmetric matrix. This relates to the choice of sign before the incompressibility constraint integral in the weak formulation (with both choices being mathematically equivalent). The incorrect choice is shown to have a hugely detrimental effect on the total computation time.