On the h-adaptive coupling of FE and BE for viscoplastic and elasto-plastic interface problems

This paper presents a posteriori error estimates for the symmetric finite element and boundary element coupling for a nonlinear interface problem: A bounded body with a viscoplastic or plastic material behaviour is surrounded by an elastic body. The nonlinearity is treated by the finite element method while large parts of the linear elastic body are approximated using the boundary element method. Based on the a posteriori error estimates we derive an algorithm for the adaptive mesh refinement of the boundary elements and the finite elements. Its implementation is documented and numerical examples are included.     

A finite element method for granular flow through a frictional boundary

A finite element method for the flow of dry granular solids through a domain involving a frictional contact boundary is formulated. The granular material is assumed as a compressible viscous-elastic–plastic continuum. Based on the principles of continuum mechanics, a complete set of equations is developed. The resulting boundary value problem is solved by the finite element method in space and by the finite difference method in time. The derivation of the finite element equations and the mathematical framework of the numerical technique are presented, together with two illustrative examples to demonstrate the validity of the technique.     

On the Description of Partially Saturated Soils by the Theory of Porous Media

In this contribution, a multi‐phase soil model based on the Theory of Porous Media (TPM) is presented. The model is fully coupled in the following constitutive phases: An elasto‐plastic or elasto‐viscoplastic solid skeleton, a materially incompressible pore‐liquid (water) and a materially compressible pore‐gas (air). The interaction of the solid skeleton and the pore‐fluids is specified by a capillary pressure‐saturation relation, whereas the mobilities of the fluid phases in the pore‐space of the solid skeleton are described by the so‐called relative permeabilities. Finally, a gravity governed initial‐boundary‐value problem solved by the FE method is presented.     

A CQM-based BEM for transient heat conduction problems in homogeneous materials and FGMs

A method for numerical solution of time-domain boundary integral formulations of transient problems governed by the heat equation is presented. The heat conduction problem is analyzed considering homogeneous and non-homogeneous media. In the case of the non-homogeneous media, the conductor material is assumed to be a functionally graded material, i.e., the material properties vary spatially according to known smooth functions. For some specific spatial variations of the material properties, the fundamental solution and the boundary integral equation of the problem are obtained thanks to a change of variables that transforms the original problem to the standard heat conduction problem for homogeneous materials. For the treatment of time-dependent terms, the convolution quadrature method is adopted to approximate numerically the integral equation of the time-domain boundary element method. In the case that the responses are required at a large number of interior points, the convolution performed to calculate them is very time consuming. It is shown that the discrete convolution of the proposed formulation can be computed by means of the fast Fourier transform technique, which considerably reduces the computational complexity. Results for some transient heat conduction examples are presented to validate the numerical techniques studied.     

Analysis of free-edge effects by boundary finite element method

The scaled boundary finite element method is a novel semi-analytical analysis technique that combines the advantages of the finite element method and the boundary element method. Only a part of the boundary of the considered domain has to be discretized but nevertheless the method is solely finite element based. The governing equations are solved in the so-called scaling direction analytically, whereas a finite element approximation of the solution is performed in the circumferential directions, which form the boundary of the considered domain. Thus, the numerical effort can be reduced considerably when handling stress concentration problems such as e.g. the free-edge effect in laminated plates. In order to analyze the free-edge effect in a semi-infinite half plane, some kinematic coupling equations have to be introduced, that not only couple the degrees of freedom on the boundary, but also within the non-discretized domain. The implementation of kinematic coupling equations within the method is presented. Finally, the efficiency of the new approach is shown in some benchmark examples. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

An approximate numerical model for radiation problems

A variety of engineering problems can be successfully solved by coupling finite element and boundary element procedures. Approximate boundary elements, which can be used when dealing with radiation problems in unlimited domains are presented. They are simple to implement and can be easily inserted in existing frontal solution packages. Numerical examples are also reported.     

Hybrid-conventional finite element for gradient-dependent plasticity

The hybrid-conventional finite element method is applied to the analysis of quasi-static, gradient-dependent elastoplastic problems in solid mechanics. The stresses within the element domain and the displacements on the boundary are simultaneously and independently approximated using Trefftz constraints, which lead to boundary integrals. The plastic multipliers are conventionally approximated with regard to C₀ continuity of the multiplier field of the gradient-dependent plasticity. The finite element formulation is derived using a Galerkin-weighted residual approach. The plastic boundary conditions are examined and plastic radiations are set to zero on the plastic boundaries. The effectiveness of the present method is demonstrated with three numerical applications.     

A posteriori finite element error estimators for indefinite elliptic boundary value problems ∗

A general construction technique is presented for a posteriori error estimators of finite element solutions of elliptic boundary value problems that satisfy a Gång inequality. The estimators are obtained by an element–by–element solution of ‘weak residual’ with or without considering element boundary residuals. There is no order restriction on the finite element spaces used for the approximate solution or the error estimation; that is, the design of the estimators is applicable in connection with either one of the h–p–, or hp– formulations of the finite element method. Under suitable assumptions it is shown that the estimators are bounded by constant multiples of the true error in a suitable norm. Some numerical results are given to demonstrate the effectiveness and efficiency of the approach.     

Analysis of a bilinear finite element for shallow shells I: Approximation of inextensional deformations

We consider a bilinear reduced-strain finite element formulation for a shallow shell model of Reissner-Naghdi type. The formulation is closely related to the facet models used in engineering practice. We estimate the error of this scheme when approximating an inextensional displacement field. We make the strong assumptions that the domain and the finite element mesh are rectangular and that the boundary conditions are periodic and the mesh uniform in one of the coordinate directions. We prove then that for sufficiently smooth fields, the convergence rate in the energy norm is of optimal order uniformly with respect to the shell thickness. In case of elliptic shell geometry the error bound is furthermore quasioptimal, whereas in parabolic and hyperbolic geometries slightly enhanced smoothness is required, except for the degenerate cases where the characteristic lines are parallel with the mesh lines. The error bound is shown to be sharp.

     

The problem of the stability of quasilinear flows with respect to perturbations of the hardening function

A formulation of the linearized boundary-value problem of the stability of a deformation process with respect to small perturbations of the hardening function (of the scalar constitutive relation of the material) is presented. The characteristic vector relations of the medium are assumed to be linear. The occurrence of rigid zones in the domain of the solid and the change in their boundaries in the perturbed motion are taken into account. A perfect rigid plastic deformation and the flow of a Newtonian fluid are considered explicitly as the basic flow. In the latter case, the equation of the asymptotic boundary of the rigid zone, which appears when there is a small variation in the yield stress and a transition to a viscoplastic material, is derived.     

A subregion DRBEM formulation for the dynamic analysis of two-dimensional cracks

The dual reciprocity boundary element method employing the step by step time integration technique is developed to analyse two-dimensional dynamic crack problems. In this method the equation of motion is expressed in boundary integral form using elastostatic fundamental solutions. In order to transform the domain integral into an equivalent boundary integral, a general radial basis function is used for the derivation of the particular solutions. The dual reciprocity boundary element method is combined with an efficient subregion boundary element method to overcome the difficulty of a singular system of algebraic equations in crack problems. Dynamic stress intensity factors are calculated using the discontinuous quarter-point elements. Several examples are presented to show the formulation details and to demonstrate the computational efficiency of the method.     

Linear B-spline finite element method for the improved Boussinesq equation

In this paper, we develop and validate a numerical procedure for solving a class of initial boundary value problems for the improved Boussinesq equation. The finite element method with linear B-spline basis functions is used to discretize the nonlinear partial differential equation in space and derive a second order system involving only ordinary derivatives. It is shown that the coefficient matrix for the second order term in this system is invertible. Consequently, for the first time, the initial boundary value problem can be reduced to an explicit initial value problem to which many accurate numerical methods are readily applicable. Various examples are presented to validate this technique and demonstrate its capacity to simulate wave splitting, wave interaction and blow-up behavior.     

计入气穴的水下爆炸作用下结构流固耦合三维数值模拟

水下爆炸在结构物面附近产生的气穴现象,严重影响水下爆炸作用下的流固耦合动响应,是舰船水下爆炸领域的难点,传统的边界元方法、有限元方法(FEM)难以解决水下爆炸气穴现象这类强非线性问题．针对此问题,计及流体中的气穴现象,考虑流体的可压缩型,忽略流体粘性,建立水下爆炸瞬态强非线性流固耦合三维数值模型,采用流体谱单元方法(SEM)和结构有限元方法求解该模型．计算结果表明:相对有限元法,谱单元法具有更高的计算精度,且谱单元解与解析解、试验值吻合良好．在此基础上,结合ABAQUS软件,分别探讨三维球壳、船体板架在水下爆炸作用下的瞬态流固耦合机理,给出气穴区域及其对水中结构物动响应的影响特征,旨在为舰船水下爆炸瞬态流固耦合问题的相关研究提供参考．     

Coupling of mixed finite element and stabilized boundary element methods for a fluid–solid interaction problem in 3D

We introduce and analyze the coupling of a mixed finite element and a boundary element for a three‐dimensional time‐harmonic fluid–solid interaction problem. We consider a formulation in which the Cauchy stress tensor and the rotation are the main variables in the elastic structure and use the usual pressure formulation in the acoustic fluid. The mixed variational formulation in the solid is completed with boundary integral equations relating the Cauchy data of the acoustic problem on the coupling interface. A crucial point in our formulation is the stabilization technique introduced by Hiptmair and coworkers to avoid the well‐known instability issue appearing in the boundary element method treatment of the exterior Helmholtz problem. The main novelty of this formulation, with respect to a previous approach, consists in reducing the computational domain to the solid media and providing a more accurate treatment of the far field effect. We show that the continuous problem is well‐posed and propose a conforming Galerkin method based on the lowest‐order Arnold–Falk–Winther mixed finite element. Finally, we prove that the numerical scheme is convergent with optimal order.Copyright © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 1211–1233, 2014     

Micromechanical Simulation of the Hall-Petch Effect with a Crystal Gradient Theory including a Grain Boundary Yield Criterion

A viscoplastic strain gradient crystal plasticity theory based on the gradient of the equivalent plastic strain ∇γ_eq is proposed. A grain boundary yield condition is introduced. The microstructural explanation of the Hall-Petch effect, accounting for notch-like stress concentrations at the grain boundary as a result of discrete slip bands, is reviewed. Periodic tensile test FEM simulation results illustrate the prediction of the numerical model. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On operator split technique for the time integration within finite strain viscoplasticity in explicit FEM

In this contribution, the operator split technique is applied to the time integration within viscoplasticity for explicit FEM. As an example, the finite strain viscoplastic material model of Shutov and Kreißig is analyzed. In the new solution scheme, some evolution equations are solved using an explicit update formula for implicit time stepping. The solution procedure is split into three steps: an elastic predictor and two viscoplastic corrector steps. Aspects of accuracy and stability of the algorithm are discussed. As shown, the proposed method is superior compared to a fully explicit integration of evolution equations. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Free-Edge Stresses around Holes in Laminates by the Boundary Finite-Element Method

In the present paper the boundary finite-element method is presented as a highly efficient technique for the numerical investigation of the free-edge stresses around a circular hole in laminates. In this method, as in the boundary element method, only the boundary needs to be discretized, whereas the element formulation in essence is finite-element based. The surface discretization provides a high numerical efficiency and requires less computation time compared to finite-element analyses. Numerical results for the concentration of interlaminar stresses at holes in composite laminates show a very good agreement with comparative finite-element calculations.     

Particle Finite Element Simulations of Cutting Processes in Manufacturing

The cutting of metals is an important process in manufacturing and challenges established methods in the field of computational mechanics. The particle finite element method (PFEM) combines the benefits of particle based methods and the standard finite element method (FEM) to account for large deformations and separation of material. In cutting simulations the workpiece is realised as a set of particles, whose boundary is detected by the α-shape method. After the boundary detection, the particles are meshed with finite elements. Since metals show a plastic behavior under large deformations, a suitable material model needs to be considered. Numerical examples show the effect of the choice of the parameter α on the cutting force. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Flow of viscoplastic materials through converging channels

Many classical rigid perfectly/plastic solutions cannot be extended to more complicated rigid plastic materials, such as rigid plastic hardening materials and rigid viscoplastic materials. The present paper reveals sources of this difficulty in the case of flow of rigid viscoplastic material through infinite converging rough wedge-shaped and axisymmetric channels. Two types of viscoplastic models without strain hardening are considered, with and with no saturation stress. The maximum friction law is adopted at the friction surface. Qualitative features of the solutions are compared to those occurring in compression of a rigid plastic layer between rough, parallel plates. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)