A dynamic variational multiscale method for viscoelasticity using linear tetrahedral elements

In this article, we develop a dynamic version of the variational multiscale (D‐VMS) stabilization for nearly/fully incompressible solid dynamics simulations of viscoelastic materials. The constitutive models considered here are based on Prony series expansions, which are rather common in the practice of finite element simulations, especially in industrial/commercial applications. Our method is based on a mixed formulation, in which the momentum equation is complemented by a pressure equation in rate form. The unknown pressure, displacement, and velocity are approximated with piecewise linear, continuous finite element functions. To prevent spurious oscillations, the pressure equation is augmented with a stabilization operator specifically designed for viscoelastic problems, in that it depends on the viscoelastic dissipation. We demonstrate the robustness, stability, and accuracy properties of the proposed method with extensive numerical tests in the case of linear and finite deformations.     

A new variational theory for linear field problems and its application to electrostatics

Abstract

A new theory, based on the concept of formally adjoint operator, is proposed to construct a variational formulation for a linear field problem, interior or exterior, with Dirichlet, Neumann, and continuity conditions. When our result is applied to the interface problem, a useful variational expression, which competes with integral equation formulation in its capability of reducing the problem dimensionality, can be obtained. The usefulness of this novel theory is demonstrated through its application to electrostatics. Three examples are also included to confirm its validity.     

Numerical analysis of an elasto‐piezoelectric problem with damage

In this paper, we analyze an algorithm for the quasistatic evolution of the mechanical state of an elasto‐piezoelectric body with damage. Both damage, caused by the development and the growth of internal microcracks, and piezoelectric effects, are included in the model. The mechanical problem is expressed as an elliptic system for the displacement field coupled with a non‐linear parabolic partial differential equation for the damage field and a linear partial differential equation for the electric potential. The variational formulation leads to a coupled system composed of two linear variational equations for the displacement field and the electric potential, and a non‐linear parabolic variational equation for the damage field. The existence of a unique weak solution is stated. Then, a fully discrete scheme is introduced by using the finite element method to approximate the spatial variable and an Euler scheme to discretize the time derivatives. Error estimates are derived on the approximate solutions, from which the linear convergence of the algorithm is deduced under suitable regularity conditions. Finally, some numerical simulations are performed, in one, two and three dimensions, to demonstrate the accuracy of the scheme and the behaviour of the solution. Copyright © 2008 John Wiley & Sons, Ltd.     

Configuration design sensitivity analysis and optimization of beam structures

 A general method for configuration design sensitivity analysis over a three-dimensional beam structure is developed based on a variational formulation of the classical beam in linear elasticity. A sensitivity formula is derived based on a variational equation for the beam structure using the material derivative concept and adjoint variable method. The formulation considers not only the shape variation in a three dimensional direction, which includes translational as well as rotational change of the beam but also the orientation angle variation of the beam's cross section. The sensitivity formula can be evaluated with generality and ease even by employing a piecewise linear design velocity field despite the fact that the bending model is a fourth order differential equation. The design sensitivity analysis is implemented using the post-processing data of a commercial code ANSYS. Several numerical examples are given to show the excellent accuracy of the method. Optimization is carried out for a tilted arch bridge and an archgrid structure to show the method's applicability. Received 29 September 2001 / Accepted 20 March 2002     

Adaptive hp-versions of boundary element methods for elastic contact problems

The variational formulation of elastic contact problems leads to variational inequalities on convex subsets. These variational inequalities are solved with the boundary element method (BEM) by making use of the Poincaré–Steklov operator. This operator can be represented in its discretized form by the Schur-complement of the dense Galerkin-matrices for the single layer potential operator, the double layer potential operator and the hypersingular integral operator. Due to the difficulties in discretizing the convex subsets involved, traditionally only the h-version is used for discretization. Recently, p- and hp-versions have been introduced for Signorini contact problems in Maischak and Stephan (Appl Numer Math, 2005) . In this paper we show convergence for the quasi-uniform hp-version of BEM for elastic contact problems, and derive a-posteriori error estimates together with error indicators for adaptive hp-algorithms. We present corresponding numerical experiments.     

A hybrid variational Allen-Cahn/ALE scheme for the coupled analysis of two-phase fluid-structure interaction

We present a partitioned iterative formulation for the modeling of fluid-structure interaction (FSI) in two-phase flows. The variational formulation consists of a stable and robust integration of three blocks of differential equations, viz, an incompressible viscous fluid, a rigid or flexible structure, and a two-phase indicator field. The fluid-fluid interface between the two phases, which may have high density and viscosity ratios, is evolved by solving the conservative phase-field Allen-Cahn equation in the arbitrary Lagrangian-Eulerian coordinates. While the Navier-Stokes equations are solved by a stabilized Petrov-Galerkin method, the conservative Allen-Cahn phase-field equation is discretized by the positivity preserving variational scheme. Fully decoupled implicit solvers for the two-phase fluid and the structure are integrated by the nonlinear iterative force correction in a staggered partitioned manner and the generalized-α method is employed for the time marching. We assess the accuracy and stability of the phase-field/ALE variational formulation for two- and three-dimensional problems involving the dynamical interaction of rigid bodies with free surface. We consider the decay test problems of increasing complexity, namely, free translational heave decay of a circular cylinder and free rotation of a rectangular barge. Through numerical experiments, we show that the proposed formulation is stable and robust for high density ratios across fluid-fluid interface and for low structure-to-fluid mass ratio with strong added-mass effects. Overall, the proposed variational formulation produces results with high accuracy and compares well with available measurements and reference numerical data. Using unstructured meshes, we demonstrate the second-order temporal accuracy of the coupled phase-field/ALE method via decay test of a circular cylinder interacting with the free surface. Finally, we demonstrate the three-dimensional phase-field FSI formulation for a practical problem of internal two-phase flow in a flexible circular pipe subjected to vortex-induced vibrations due to external fluid flow.     

Exterior stable domain segmentation integral equation method for scattering problems

This paper is concerned with the development of an exterior domain segmentation method for the solution of two- or three-dimensional time-harmonic scattering problems in acoustic media. This method, based on a variational localized, symmetric, boundary integral equation formulation leads, upon discretization, to a sparse system of algebraic equations whose solution requires only O(N) multiplications, where N is the number of unknown nodal pressures on the scatterer surface. The new procedure is analogous to the one developed recently¹ except that in the present formulation we avoid completely the use of the hypersingular operator, thereby reducing the computational complexity. Numerical experiments for a rigid circular cylindrical scatterer subjected to a plane incident wave serve to assess its accuracy for normalized wave numbers, ka, ranging from 0 to 30, both directly on the scatterer and in the far field, and to confirm that, contrary to standard boundary integral equation formulations, the present procedure is valid for critical frequencies.     

A mixed-variable continuously deforming finite element method for parabolic evolution problems. Part I: The variational formulation for a single evolution equation

The objective of the research presented here was to develop a generic adaptive computational method for porous media evolution problems that involve coupled heat flow, fluid flow and species transport processes with sharply defined phase-change interfaces. In this paper we examine the general least squares variational approach and develop the conceptual framework for a rate least squares variational formulation of a continuously deforming mixed variable finite element method for solving highly non-linear time-dependent partial differential equations. In Part II of this paper¹ we extend the formulation given here for a single evolution equation to a system of coupled evolution equations. In Part III² we discuss in detail the numerical procedures that were implemented in a computer program and present several numerical examples that demonstrate the performance of this computational method.     

Solution of the time‐harmonic viscoelastic inverse problem with interior data in two dimensions

We consider the problem of determining the distribution of the complex‐valued shear modulus for an incompressible linear viscoelastic material undergoing infinitesimal time‐harmonic deformation, given the knowledge of the displacement field in its interior. In particular, we focus on the two‐dimensional problems of anti‐plane shear and plane stress. These problems are motivated by applications in biomechanical imaging, where the material modulus distributions are used to detect and/or diagnose cancerous tumors. We analyze the well‐posedness of the strong form of these problems and conclude that for the solution to exist, the measured displacement field is required to satisfy rather restrictive compatibility conditions. We propose a weak, or a variational formulation, and prove the existence and uniqueness of solutions under milder conditions on measured data. This formulation is derived by weighting the original PDE for the shear modulus by the adjoint operator acting on the complex‐conjugate of the weighting functions. For this reason, we refer to it as the complex adjoint weighted equation (CAWE). We consider a straightforward finite element discretization of these equations with total variation regularization, and test its performance with synthetically generated and experimentally measured data. We find that the CAWE method is, in general, less diffusive than a corresponding least squares solution, and that the total variation regularization significantly improves its performance in the presence of noise. Copyright © 2012 John Wiley & Sons, Ltd.     

Finite element procedures for the nonlinear dynamic stability analysis of arbitrary shell structures

The objective is the development of numerical algorithms for the dynamic stability analysis of strongly nonlinear shell structures subjected, in particular, to parametric excitations. The finite-element discretization is achieved by displacement models of high accuracy. The basis for the stability analysis is Ljapunow's first approximation equation obtained from a finite-rotation shell theory by a variational method. The operator formulation used for this purpose shows the mathematical requirements imposed on consistent formulations. In close connection with Floquet's theory, a semi-analytical criterion is finally given for the stability analysis of parametric instability phenomena. The numerical results presented demonstrate the efficiency of the numerical algorithms.     

On the a priori and a posteriori error analysis of a two‐fold saddle‐point approach for nonlinear incompressible elasticity

In this paper, we reconsider the a priori and a posteriori error analysis of a new mixed finite element method for nonlinear incompressible elasticity with mixed boundary conditions. The approach, being based only on the fact that the resulting variational formulation becomes a two‐fold saddle‐point operator equation, simplifies the analysis and improves the results provided recently in a previous work. Thus, a well‐known generalization of the classical Babu?ka–Brezzi theory is applied to show the well‐posedness of the continuous and discrete formulations, and to derive the corresponding a priori error estimate. In particular, enriched PEERS subspaces are required for the solvability and stability of the associated Galerkin scheme. In addition, we use the Ritz projection operator to obtain a new reliable and quasi‐efficient a posteriori error estimate. Finally, several numerical results illustrating the good performance of the associated adaptive algorithm are presented. Copyright © 2006 John Wiley & Sons, Ltd.     

A class of preconditioned conjugate gradient methods for the solution of a mixed finite element discretization of the biharmonic operator

The homogeneous Dirichlet problem for the biharmonic operator is solved as the variational formulation of two coupled second-order equations. The discretization by a mixed finite element model results in a set of linear equations whose coefficient matrix is sparse, symmetric but indefinite. We describe a class of preconditioned conjugate gradient methods for the numerical solution of this linear system. The precondition matrices correspond to incomplete factorizations of the coefficient matrix. The numerical results show a low computational complexity in both number of computer operations and demand of storage.     

An error‐estimate‐free and remapping‐free variational mesh refinement and coarsening method for dissipative solids at finite strains

A variational h‐adaptive finite element formulation is proposed. The distinguishing feature of this method is that mesh refinement and coarsening are governed by the same minimization principle characterizing the underlying physical problem. Hence, no error estimates are invoked at any stage of the adaption procedure. As a consequence, linearity of the problem and a corresponding Hilbert‐space functional framework are not required and the proposed formulation can be applied to highly non‐linear phenomena. The basic strategy is to refine (respectively, unrefine) the spatial discretization locally if such refinement (respectively, unrefinement) results in a sufficiently large reduction (respectively, sufficiently small increase) in the energy. This strategy leads to an adaption algorithm having O(N) complexity. Local refinement is effected by edge‐bisection and local unrefinement by the deletion of terminal vertices. Dissipation is accounted for within a time‐discretized variational framework resulting in an incremental potential energy. In addition, the entire hierarchy of successive refinements is stored and the internal state of parent elements is updated so that no mesh‐transfer operator is required upon unrefinement. The versatility and robustness of the resulting variational adaptive finite element formulation is illustrated by means of selected numerical examples. Copyright © 2008 John Wiley & Sons, Ltd.     

A new boundary-type finite element for 2-D- and 3-D-elastic structures

In this paper we discuss the theoretical and numerical formulation of 3-D Trefitz elements. Starting from the variational principle with the so-called hybrid stress method, the trial functions for the stresses have to fulfil the Beltrami equations, which means also the compatibility equations for the strains. The divergence theorem can be applied, and one arrives at a pure boundary formulation in the sense of the Trefftz method. Besides the resulting variational formulation, different regularizations of the interelement conditions are investigated by numerical tests. Two examples show the numerical efficiency of the derived elements. First, a geometric linear 3-D example is presented to show the effects on distorted element meshes. The third example shows the geometrically non-linear analysis of a shallow cylindrical shell segment under a singe load.     

A mixed-variable continuously deforming finite element method for parabolic evolution problems. Part III: Numerical implementation and computational results

In Part I of this paper,¹ the conceptual framework of a rate variational least squares formulation of a continuously deforming mixed-variable finite element method was presented for solving a single evolution equation. In Part II² a system of ordinary differential equations with respect to time was derived for solving a system of three coupled evolution equations by the deforming grid mixed-variable least squares rate variational finite element method. The system of evolution equations describes the coupled heat flow, fluid flow and trace species transport in porous media under conditions when the flow velocities and constituent phase transitions induce sharp fronts in the solution domain. In this paper, we present the method we have adopted to integrate with respect to time the resulting spatially discretized system of non-linear ordinary differential equations. Next, we present computational results obtained using the code in which this deforming mixed finite element method was implemented. Because several features of the formulation are novel and have not been previously attempted, the problems were selected to exercise these features with the objective of demonstrating that the formulation is correct and that the numerical procedures adopted converge to the correct solutions.     

基于Allen-Cahn方程图像修复的算子分裂方法（英）

本文提出了一种基于Allen-Cahn方程图像修复的算子分裂方法．其核心思想是利用算子分裂方法将原问题分解为一个线性方程和一个非线性方程,线性方程使用有限差分Crank-Nicolson格式进行离散,非线性方程利用解析方法进行求解,因此时间和空间都能达到二阶精度．由于该方法只作用于图像需要修复的区域,而其余区域的像素值与原始图像的保持一样,可以大大提高计算效率．合成图像和真实图像的数值实验验证了该算法的正确性和有效性．     

Regular hybrid boundary node method for biharmonic problems

The regular hybrid boundary node method (RHBNM) is a new technique for the numerical solutions of the boundary value problems. By coupling the moving least squares (MLS) approximation with a modified functional, the RHBNM retains the meshless attribute and the reduced dimensionality advantage. Besides, since the source points of the fundamental solutions are located outside the domain, ‘boundary layer effect’ is also avoided. However, an initial restriction of the present method is that it is only suitable for the problems which the governing differential equation is in second order.Now, a new variational formulation for the RHBNM is presented further to solve the biharmonic problems, in which the governing differential equation is in fourth order. The modified variational functional is applied to form the discrete equations of the RHBNM. The MLS is employed to approximate the boundary variables, while the domain variables are interpolated by a linear combination of fundamental solutions of both the biharmonic equation and Laplace’s equation. Numerical examples for some biharmonic problems show that the high accuracy with a small node number is achievable. Furthermore, the computation parameters have been studied. They can be chosen in a wide range and have little influence on the results. It is shown that the present method is effective and can be widely applied in practical engineering.     

Perturbed incomplete ILU preconditioner for efficient solution of electric field integral equations

To efficiently solve large, dense, complex linear systems that arise in the electric field integral equation (EFIE) formulation of electromagnetic scattering problems, a new modified incomplete LU (ILU) preconditioner is developed and used in the context of the generalised minimal residual iterative method accelerated with the multilevel fast multipole method. The key idea is to perturb the near-field impedance matrix of EFIE with the principle value term of the magnetic field integral equation operator before constructing ILU preconditioners. Numerical experiments indicate that this new perturbation technique is very effective with the ILU preconditioner and the resulted ILU preconditioner can reduce both the iteration number and the computational time substantially.     

A Nonlocal Operator Method for Partial Differential Equations with Application to Electromagnetic Waveguide Problem

A novel nonlocal operator theory based on the variational principle is proposed for the solution of partial differential equations. Common differential operators as well as the variational forms are defined within the context of nonlocal operators. The present nonlocal formulation allows the assembling of the tangent stiffness matrix with ease and simplicity, which is necessary for the eigenvalue analysis such as the waveguide problem. The present formulation is applied to solve the differential electromagnetic vector wave equations based on electric fields. The governing equations are converted into nonlocal integral form. An hourglass energy functional is introduced for the elimination of zero-energy modes. Finally, the proposed method is validated by testing three classical benchmark problems.     

Numerical modelling of non‐linear electroelasticity

The numerical modelling of non‐linear electroelasticity is presented in this work. Based on well‐established basic equations of non‐linear electroelasticity a variational formulation is built and the finite element method is employed to solve the non‐linear electro‐mechanical coupling problem. Numerical examples are presented to show the accuracy of the implemented formulation. Copyright © 2006 John Wiley & Sons, Ltd.