A simple,stable, and accurate linear tetrahedral finite element for transient,nearly, and fully incompressible solid dynamics: a dynamic variational multiscale approach

We propose a new approach for the stabilization of linear tetrahedral finite elements in the case of nearly incompressible transient solid dynamics computations. Our method is based on a mixed formulation, in which the momentum equation is complemented by a rate equation for the evolution of the pressure field, approximated with piecewise linear, continuous finite element functions. The pressure equation is stabilized to prevent spurious pressure oscillations in computations. Incidentally, it is also shown that many stabilized methods previously developed for the static case do not generalize easily to transient dynamics. Extensive tests in the context of linear and nonlinear elasticity are used to corroborate the claim that the proposed method is robust, stable, and accurate. Copyright © 2015 John Wiley & Sons, Ltd.     

Stabilized low‐order finite elements for strongly coupled poromechanical problems

Finite element solutions of poromechanical problems often exhibit oscillating pore pressures in the limits of low permeability, fast loading rates, coarse meshes, and/or small time step sizes. To suppress completely the pore pressure oscillations, a stabilized finite element scheme with a better performance on monotonicity is proposed for modeling compressible fluid‐saturated porous media. This method, based on the polynomial pressure projection technique, allows the use of linear equal‐order interpolation for both displacement and pore pressure fields, which is more straightforward for both code development and maintenance compared to others. By employing the discrete maximum principle, a proper stabilization parameter is deduced, which is efficient to guarantee the monotonicity and optimal in theory in the 1‐dimensional case. An appealing feature of the method is that the stabilization parameter is evaluated in terms of the properties of porous material only, while no mesh or time step size is involved. Through comparing the numerical simulations with the analytical benchmarks, the efficiency of the proposed stabilization scheme is confirmed.     

粘弹流动的间断有限元模拟

针对传统有限元法求解Oldroyd-B本构方程时需加入稳定化方案的缺点,本文基于非结构网格给出了统一间断有限元求解框架.该框架包含采用IPDG(interior penalty discontinuous Galerkin)求解质量方程和动量方程,与采用RKDG(RungeKutta discontinuous Galerkin)求解本构方程这两个核心.数值结果表明:该方法在求解Oldroyd-B本构方程时无需加入稳定化方案,实施比有限元法简便,且具有较高的计算精度,可有效地模拟包含应力奇异点的复杂粘弹流动问题,进而揭示非牛顿粘弹流动的基本特征.     

Numerical modeling of micro- and macro-behavior of viscoelastic porous materials

This paper presents a numerical method for solving the two-dimensional problem of a polygonal linear viscoelastic domain containing an arbitrary number of non-overlapping circular holes of arbitrary sizes. The solution of the problem is based on the use of the correspondence principle. The governing equation for the problem in the Laplace domain is a complex hypersingular boundary integral equation written in terms of the unknown transformed displacements on the boundaries of the holes and the exterior boundaries of the finite body. No specific physical model is involved in the governing equation, which means that the method is capable of handling a variety of viscoelastic models. A truncated complex Fourier series with coefficients dependent on the transform parameter is used to approximate the unknown transformed displacements on the boundaries of the holes. A truncated complex series of Chebyshev polynomials with coefficients dependent on the transform parameter is used to approximate the unknown transformed displacements on the straight boundaries of the finite body. A system of linear algebraic equations is formed using the overspecification method. The viscoelastic stresses and displacements are calculated through the viscoelastic analogs of the Kolosov–Muskhelishvili potentials, and an analytical inverse Laplace transform is used to provide the time domain solution. Using the concept of representative volume, the effective viscoelastic properties of an equivalent homogeneous material are then found directly from the corresponding constitutive equations for the average field values. Several examples are given to demonstrate the accuracy of the method. The results for the stresses and displacements are compared with the numerical solutions obtained by commercial finite element software (ANSYS). The results for the effective properties are compared with those obtained with the self-consistent and Mori–Tanaka schemes.     

The mixed finite element method for the quasi‐static and dynamic analysis of viscoelastic timoshenko beams

The quasi‐static and dynamic responses of a linear viscoelastic Timoshenko beam on Winkler foundation are studied numerically by using the hybrid Laplace–Carson and finite element method. In this analysis the field equation for viscoelastic material is used. In the transformed Laplace–Carson space two new functionals have been constructed for viscoelastic Timoshenko beams through a systematic procedure based on the Gâteaux differential. These functionals have six and two independent variables respectively. Two mixed finite element formulations are obtained; TB12 and TB4. For the inverse transform Schapery and Fourier methods are used. The numerical results for quasi‐static and dynamic responses of several visco‐elastic models are presented. Copyright © 1999 John Wiley & Sons, Ltd.     

Stabilized finite element method for viscoplastic flow: formulation with state variable evolution

A stabilized, mixed finite element formulation for modelling viscoplastic flow, which can be used to model approximately steady‐state metal‐forming processes, is presented. The mixed formulation is expressed in terms of the velocity, pressure and state variable fields, where the state variable is used to describe the evolution of the material's resistance to plastic flow. The resulting system of equations has two sources of well‐known instabilities, one due to the incompressibility constraint and one due to the convection‐type state variable equation. Both of these instabilities are handled by adding mesh‐dependent stabilization terms, which are functions of the Euler–Lagrange equations, to the usual Galerkin method. Linearization of the weak form is derived to enable a Newton–Raphson implementation into an object‐oriented finite element framework. A progressive solution strategy is used for improving convergence for highly non‐linear material behaviour, typical for metals. Numerical experiments using the stabilization method with hierarchic shape functions for the velocity, pressure and state variable fields in viscoplastic flow and metal‐forming problems show that the stabilized finite element method is effective and efficient for non‐linear steady forming problems. Finally, the results are discussed and conclusions are inferred. Copyright © 2002 John Wiley & Sons, Ltd.     

Numerical stabilization of Biot's consolidation model by a perturbation on the flow equation

In this paper a stabilized finite element scheme for the poroelasticity equations is proposed. This method, based on the perturbation of the flow equation, allows us to use continuous piecewise linear approximation spaces for both displacements and pressure, obtaining solutions without oscillations independently of the chosen discretization parameters. The perturbation term depends on a parameter which is established in terms of the mesh size and the properties of the material. In the one‐dimensional case, this parameter is shown to be optimal. Some numerical experiments are presented indicating the efficiency of the proposed stabilization technique. Copyright © 2008 John Wiley & Sons, Ltd.     

An assumed‐gradient finite element method for the level set equation

The level set equation is a non‐linear advection equation, and standard finite‐element and finite‐difference strategies typically employ spatial stabilization techniques to suppress spurious oscillations in the numerical solution. We recast the level set equation in a simpler form by assuming that the level set function remains a signed distance to the front/interface being captured. As with the original level set equation, the use of an extensional velocity helps maintain this signed‐distance function. For some interface‐evolution problems, this approach reduces the original level set equation to an ordinary differential equation that is almost trivial to solve. Further, we find that sufficient accuracy is available through a standard Galerkin formulation without any stabilization or discontinuity‐capturing terms. Several numerical experiments are conducted to assess the ability of the proposed assumed‐gradient level set method to capture the correct solution, particularly in the presence of discontinuities in the extensional velocity or level‐set gradient. We examine the convergence properties of the method and its performance in problems where the simplified level set equation takes the form of a Hamilton–Jacobi equation with convex/non‐convex Hamiltonian. Importantly, discretizations based on structured and unstructured finite‐element meshes of bilinear quadrilateral and linear triangular elements are shown to perform equally well. Copyright © 2005 John Wiley & Sons, Ltd.     

Application of the control volume mixed finite element method to a triangular discretization

A two‐dimensional control volume mixed finite element method is applied to the elliptic equation. Discretization of the computational domain is based in triangular elements. Shape functions and test functions are formulated on the basis of an equilateral reference triangle with unit edges. A pressure support based on the linear interpolation of elemental edge pressures is used in this formulation. Comparisons are made between results from the standard mixed finite element method and this control volume mixed finite element method. Published 2011. This article is a US Government work and is in the public domain in the USA.     

Modelling of the crack growth initiation in viscoelastic media by the Gθv-integral

In this paper, the effects of viscoelastic characteristics on the creep crack growth initiation are studied by developing a new path-independent integral Gθ_v which allows us to compute the energy release rate with the finite element method. The originality of this approach is the perfect uncoupling between the viscous dissipation and the free energy which drives the crack propagation and the crack growth initiation. Coupled with an explicit finite element formulation of the linear viscoelastic behavior, this integral allows us to simulate accurate crack growth initiation.     

A generalized layerwise finite element for multi-layer damping treatments

This paper presents a 4-node facet type quadrangular shell finite element, based on a layerwise theory, developed for dynamic modelling of laminated structures with viscoelastic damping layers. The bending stiffness of the facet shell element is based on the Reissner–Mindlin assumptions and the plate theory is enriched with a shear locking protection adopting the MITC approach. The membrane component is corrected by using incompatible quadratic modes and the drilling degrees of freedom are introduced through a fictitious stiffness stabilization matrix. Linear static tests, using several pathological tests, showed good and convergent results. Dynamic analysis evaluation is provided by using two eigenproblems with exact analytical solution, as well as a conical sandwich shell with a closed-form analytical solution and a semi-analytical ring finite element solution. The applicability of the proposed finite element to viscoelastic core sandwich plates is assessed through experimental validation.     

A stabilization technique for the regularization of nearly singular extended finite elements

In this contribution a simple, robust and efficient stabilization technique for extended finite element (XFEM) simulations is presented. It is useful for arbitrary crack geometries in two or three dimensions that may lead to very bad condition numbers of the global stiffness matrix or even ill-conditioning of the equation system. The method is based on an eigenvalue decomposition of the element stiffness matrix of elements that only possess enriched nodes. Physically meaningful zero eigenmodes as well as enrichment scheme dependent numerically reasonable zero eigenmodes are filtered out. The remaining subspace is stabilized depending on the magnitude of the respective eigenvalues. One of the main advantages is the fact that neither the equation solvers need to be changed nor the solution method is restricted. The efficiency and robustness of the method is demonstrated in numerous examples for 2D and 3D fracture mechanics.     

VPF粘性介质粘弹性本构关系研究

为研究粘性介质的力学性能对板材变形的影响,通过剪切蠕变-回复和松弛试验分析了甲基乙烯基粘性介质的流变性能.实验结果表明粘性介质可以简化为线性粘弹性材料.建立了粘性介质的积分型粘弹性本构方程,并结合剪切蠕变-回复过程的有限元分析确定了方程中的材料参数.利用该本构方程对粘性介质压力胀形过程进行了有限元分析,模拟结果与试验结果对比表明,所建立的本构方程可以较好的预测板材的变形过程.     

粘弹性材料的ADF模型研究

采用GA-BFGS混合遗传算法建立了一套确定粘弹性模型参数的方法,对ADF模型进行参数拟合,并与标准流变模型,GHM模型及试验结果进行了比较。基于ADF模型的本构方程和控制方程,建立了粘弹性自由层阻尼悬臂梁的有限元方程。算例表明,本文确定模型参数的方法是有效的;同时通过对自由层悬臂梁的有限元模态分析,表明ADF模型在描述粘弹性材料本构关系中具有结构简单,计算精确,方便和有限元相结合的特点。     

Stabilized finite element methods for vertically averaged multiphase flow for carbon sequestration

A computationally efficient numerical model that describes carbon sequestration in deep saline aquifers is presented. The model is based on the multiphase flow and vertically averaged mass balance equations, requiring the solution of two partial differential equations – a pressure equation and a saturation equation. The saturation equation is a nonlinear advective equation for which the application of Galerkin finite element method (FEM) can lead to non‐physical oscillations in the solution. In this article, we extend three stabilized FEM formulations, which were developed for uncoupled systems, to the governing nonlinear coupled PDEs. The methods developed are based on the streamline upwind, the streamline upwind/Petrov–Galerkin and the least squares FEM. Two sequential solution schemes are developed: a single step and a predictor–corrector. The range of Courant numbers yielding smooth and oscillation‐free solutions is investigated for each method. The useful range of Courant numbers found depends upon both the sequential scheme (single step vs predictor–corrector) and also the time integration method used (forward Euler, backward Euler or Crank–Nicolson). For complex problems such as when two plumes meet, only the SU stabilization with an amplified stabilization parameter gives satisfactory results when large time steps are used. Copyright © 2016 John Wiley & Sons, Ltd.     

基于CFD/CSD耦合的结构几何非线性静气动弹性数值方法研究

柔性飞行器在气动力作用下会发生大变形,产生结构几何非线性,线性小变形方法难以获得准确的气动弹性分析结果。基于RANs的三维N-S流场控制方程耦合非线性结构静力学方程时域分析方法,用于考虑结构几何非线性的静气动弹性分析。该方法在结构静力学方程求解上采用非线性增量有限元方法进行迭代求解,考虑结构刚度矩阵随结构位形的变化,采用径向基函数方法实现气动/结构界面的数据交换和动网格变形。在建立某型宽体客机复材机翼三维有限元模型的基础上,对其静气动弹性进行了数值仿真,分析了线性结构和考虑结构几何非线性的结构在静气动弹性作用下翼面扭转、展向位移、垂向位移以及升力系数等物理量。算例结果表明,与线性结果相比,非线性结构由于结构几何非线性的影响,在展向和垂向变形上两者存在显著差异。为准确进行柔性结构的气动弹性分析,必须考虑结构几何非线性的影响。     

Elastoplasticity with linear tetrahedral elements: A variational multiscale method

We present a computational framework for the simulation of J₂‐elastic/plastic materials in complex geometries based on simple piecewise linear finite elements on tetrahedral grids. We avoid spurious numerical instabilities by means of a specific stabilization method of the variational multiscale kind. Specifically, we introduce the concept of subgrid‐scale displacements, velocities, and pressures, approximated as functions of the governing equation residuals. The subgrid‐scale displacements/velocities are scaled using an effective (tangent) elastoplastic shear modulus, and we demonstrate the beneficial effects of introducing a subgrid‐scale pressure in the plastic regime. We provide proofs of stability and convergence of the proposed algorithms. These methods are initially presented in the context of static computations and then extended to the case of dynamics, where we demonstrate that, in general, naïve extensions of stabilized methods developed initially for static computations seem not effective. We conclude by proposing a dynamic version of the stabilizing mechanisms, which obviates this problematic issue. In its final form, the proposed approach is simple and efficient, as it requires only minimal additional computational and storage cost with respect to a standard finite element relying on a piecewise linear approximation of the displacement field.     

Numerical operational methods for time-dependent linear problems

A general and systematic discussion on the use of the operational method of Laplace transform for numerically solving complex time-dependent linear problems is presented. Application of Laplace transform with respect to time on the governing differential equations as well as the boundary and initial conditions of the problem reduces it to one independent of time, which is solved in the transform domain by any convenient numerical technique, such as the finite element method, the finite difference method or the boundary integral equation method. Finally, the time domain solution is obtained by a numerical inversion of the transformed solution. Eight existing methods of numerical inversion of the Laplace transform are systematically discussed with respect to their use, range of applicability, accuracy and computational efficiency on the basis of some framework vibration problems. Other applications of the Laplace transform method in conjunction with the finite element method or the boundary integral equation method in the areas of earthquake dynamic response of frameworks, thermaliy induced beam vibrations, forced vibrations of cylindrical shells, dynamic stress concentrations around holes in plates and viscoelastic stress analysis are also briefly described to demonstrate the generality and advantages of the method against other known methods.     

A Galerkin/least‐squares finite element formulation for consolidation

A Galerkin/least‐squares (GLS) finite element formulation for problem of consolidation of fully saturated two‐phase media is presented. The elimination of spurious pressure oscillations appearing at the early stage of consolidation for standard Galerkin finite elements with equal interpolation order for both displacements and pressures is the goal of the approach. It will be shown that the least‐squares term, based exclusively on the residuum of the fluid flow continuity equation, added to the standard Galerkin formulation enhances its stability and can fully eliminate pressure oscillations. A reasonably simple framework designed for derivation of one‐dimensional as well as multi‐dimensional estimates of the stabilization factor is proposed and then verified. The formulation is validated on one‐dimensional and then on two‐dimensional, linear and non‐linear test problems. The effect of the fluid incompressibility as well as compressibility will be taken into account and investigated. Copyright © 2001 John Wiley & Sons Ltd.     

Finite element formulation of viscoelastic sandwich beams using fractional derivative operators

This paper presents a finite element formulation for transient dynamic analysis of sandwich beams with embedded viscoelastic material using fractional derivative constitutive equations. The sandwich configuration is composed of a viscoelastic core (based on Timoshenko theory) sandwiched between elastic faces (based on Euler–Bernoulli assumptions). The viscoelastic model used to describe the behavior of the core is a four-parameter fractional derivative model. Concerning the parameter identification, a strategy to estimate the fractional order of the time derivative and the relaxation time is outlined. Curve-fitting aspects are focused, showing a good agreement with experimental data. In order to implement the viscoelastic model into the finite element formulation, the Grünwald definition of the fractional operator is employed. To solve the equation of motion, a direct time integration method based on the implicit Newmark scheme is used. One of the particularities of the proposed algorithm lies in the storage of displacement history only, reducing considerably the numerical efforts related to the non-locality of fractional operators. After validations, numerical applications are presented in order to analyze truncation effects (fading memory phenomena) and solution convergence aspects.