Non-symmetric Formulations Research Articles

Variational formulation of Cahn–Hilliard-type diffusion coupled with crystal plasticity

This article presents a relatively general framework for the description of the Cahn–Hilliard-type diffusion in solids undergoing infinitesimal elastic and plastic deformations. The coupled chemo-mechanical problem, characterised by phenomena such as phase segregation, microstructure coarsening and swelling, is treated using the variational framework which is governed by continuous-time, discrete-time and discrete-space–time incremental variational principles. It is shown that the governing equations of the coupled problem can be derived as Euler equations of minimisation and saddle point principles. A point of departure from the existing works is the coupling of crystal plasticity to the problem of diffusion and optimising the potential with respect to the plastic variables such that they are solved locally at the integration points. This is done using a return map algorithm which results in a reduced global problem. The variational framework results in a system of symmetric non-linear algebraic equations that are solved by Newton–Raphson-type iterative methods. This is a novel and attractive feature with respect to numerical implementation, as models resulting from the proposed variational framework are computationally less expensive in comparison with non-symmetric formulations. The numerical simulations presented at the end predict the applicability of models resulting from the proposed variational framework for multiple scenarios.

Boundary Integral Formulation of Frictionless Contact Problems Based on an Energetic Approach and Its Numerical Implementation by the Collocation BEM

A unified methodology to solve problems of frictionless unilateral contact as well as adhesive contact between linear elastic solids is presented. This methodology is based on energetic principles and is casted to a minimization problem of the total potential energy. Appropriate boundary integral forms of the energy are defined and the quadratic problem form of the contact problem is proposed. The problem is solved by the collocation boundary element method (BEM). To solve the quadratic problem two algorithms are developed, both being variants of the well-known conjugate gradient algorithm. The difference between them is given by an explicit construction or not of the quadratic-problem matrix. This matrix has the same physical meaning as the stiffness matrix commonly used in the context of the finite element method (FEM). Both symmetric and non-symmetric formulations of this matrix are presented and discussed, showing that the non-symmetric one provides more accurate results. The present procedure, in addition to its interest by itself, can also be extended to problems where dissipative phenomena take place such as friction, damage and plasticity. Elements of the numerical implementation are briefly presented and the numerical solution of some standard problems of frictionless contact are given and compared to those obtained by other well-known BEM and FEM procedures for contact problems.

On the DPG method for Signorini problems

We derive and analyze discontinuous Petrov-Galerkin methods with optimal test functions for Signorini-type problems as a prototype of a variational inequality of the first kind. We present different symmetric and non-symmetric formulations where optimal test functions are only used for the PDE part of the problem, not the boundary conditions. For the symmetric case and lowest order approximations, we provide a simple a posteriori error estimate. In a second part, we apply our technique to the singularly perturbed case of reaction dominated diffusion. Numerical results show the performance of our method and, in particular, its robustness in the singularly perturbed case.

Mixed shell element for static and buckling analysis of variable angle tow composite plates

A mixed quadrilateral 3D finite element, obtained from the Hellinger–Reissner functional, is presented for linear static and buckling analyses of variable-angle tow (VAT) composite plates. Variable-angle tows describe curvilinear fiber paths within composite laminae and are a promising technology for tailoring the buckling and post-buckling capability of plates. Due to the variable stiffness across the planform of the VAT plates, pre-buckling stresses can be tailored and redistributed towards supported edges, thereby greatly improving the buckling load. A linear mixed element called MISS-4 is used as starting point for this work. The element presents a self-equilibrated and isostatic state of stress. The kinematics lead to element compatibility matrix calculations based solely on the interpolation along element edges. The drilling rotations do not require penalty functions or non-symmetric formulations, thus avoiding spurious energy modes. The buckling analysis is reliably performed via a co-rotational formulation. In this work VAT plates with linear fiber angle variation in one direction, and constant stiffness properties in the orthogonal direction are studied. Numerical examples of VAT plates subjected to different loads and boundary conditions are investigated herein. The convergence of displacements, stress resultants and buckling loads are presented, and comparisons with numerical results, obtained using the S4R finite element of Abaqus and the pseudo-spectral Differential Quadrature Method, are shown.

Transformation cloaking and radial approximations for flexural waves in elastic plates

It is known that design of elastic cloaks is much more challenging than that of acoustic cloaks, cloaks of electromagnetic waves or scalar problems of anti-plane shear. In this paper, we address fully the fourth-order problem and develop a model of a broadband invisibility cloak for channelling flexural waves in thin plates around finite inclusions. We also discuss an option to employ efficiently an elastic pre-stress and body forces to achieve such a result. An asymptotic derivation provides a rigorous link between the model in question and elastic wave propagation in thin solids. This is discussed in detail to show connection with non-symmetric formulations in vector elasticity studied in earlier work.

A mixed isostatic 24 dof element for static and buckling analysis of laminated folded plates

A mixed, quadrilateral, 3D plate element is proposed for static linear and buckling analysis of folded laminated composite structures. Numerical results show accuracy, comparing to analytical and numerical references, and convergence rate h2 measured using an s-norm. These characteristics are due to the self-equilibrated, isostatic state of stress in the element, and to the element kinematics leading to element compliance and compatibility matrix calculations based solely on the interpolation along the element edges. For folded plates, the drilling rotations do not require penalty functions or non-symmetric formulations, thus avoiding spurious energy modes. Buckling analysis is achieved by a corotational formulation, which is possible thanks to the accuracy of rotation approximations. Benchmarking for laminated composite plates includes convergence of displacements and stress-resultants, global error measures, and comparison with literature results.

A comparison between a symmetric and a non-symmetric Galerkin finite element—boundary integral equation coupling for the two-dimensional exterior Stokes problem

Symmetric and non-symmetric Galerkin formulations are presented for the coupling of a finite element modelled interior region to a boundary integral supported exterior region for the two-dimensional steady state exterior Stokes problem. Both single and double-layer hydrodynamic potentials are used allowing a well conditioned symmetric matrix structure for the entire interior–exterior, velocity–pressure system when the exterior velocity boundary integral equation (VBIE) is augmented by a traction boundary integral equation (TBIE) with the pressure determined everywhere purely through the imposition of the divergence-free velocity condition. Corresponding non-symmetric formulations are obtained by additionally discretizing an associated pressure boundary integral equation (PBIE), where the associated kernel functions have singularities an order higher than in the VBIE, with a simple regularization of the new hyper-singular pressure kernel. Comparable solution convergence with mesh refinement for the symmetric and non-symmetric schemes is shown for stabilized and mixed velocity–pressure conforming finite element pairs using Lagrangian shape functions.

Stabilized discontinuous finite element approximations for Stokes equations

In this paper, we derive two stabilized discontinuous finite element formulations, symmetric and nonsymmetric, for the Stokes equations and the equations of the linear elasticity for almost incompressible materials. These methods are derived via stabilization of a saddle point system where the continuity of the normal and tangential components of the velocity/displacements are imposed in a weak sense via Lagrange multipliers. For both methods, almost all reasonable pair of discontinuous finite element spaces can be used to approximate the velocity and the pressure. Optimal error estimate for the approximation of both the velocity of the symmetric formulation and pressure in L 2 norm are obtained, as well as one in a mesh-dependent norm for the velocity in both symmetric and nonsymmetric formulations.

Geometrically non-linear and consistently linearized embedded strong discontinuity models for 3D problems with an application to the dissection analysis of soft biological tissues

Three different finite element formulations with embedded strong discontinuities are derived on the basis of the enhanced assumed strain method. According to the work by Jirásek and Zimmermann [Int. J. Numer. Methods Engrg. 50 (2001) 1269] they are referred to as statically optimal symmetric (SOS), kinematically optimal symmetric (KOS) and statically and kinematically optimal non-symmetric (SKON) formulations. The effect of the discontinuities are characterized by additional degrees of freedom on the element level. Modifications to the standard KOS and SKON formulations are proposed in order to achieve consistency with the employed type of a three-field Hu–Washizu principle under mode-I condition. Under this condition the formulation satisfies the internal compatibility at the discontinuity, i.e. the relation between the stress in the bulk material and the traction across the discontinuity surface, which is not the case for the classical KOS formulation. We propose a suitable explicit expression for a transversely isotropic traction law in form of a displacement–energy function and assume that softening phenomena in the cohesive zone are modeled by a damage law, which depends on the maximum gap displacement of the deformation path. A linearization of all quantities, which are related to the non-linear problem, leads to new closed form expressions. In particular, we focus attention on the linearization of the cohesive traction vector. The associated element residua and stiffness matrices are provided. Standard static condensation of the internal degree of freedom leads to a generalized displacement model. A comparative study of the modified formulations, carried out by means of two numerical examples, show the performance of the individual approach. We employ constant-strain tetrahedral elements with a single discontinuity embedded. Among the known stress locking phenomena associated with the SOS formulation, we recognized that the (non-symmetric) SKON formulation was not able to provide meaningful results for the dissection process of an arterial layer in three-dimensions on distorted meshes. For both numerical examples the (symmetric) KOS formulation seems to be most suitable for representing the embedded discontinuities.

Further research on the performance of consistent mass matrices using BEM for symmetric/nonsymmetric formulations

This paper deals with the performance of consistent mass matrices for the 2-D scalar wave propagation problem using the Boundary Element Method (BEM), and proposes a new global functional set of base functions capable of avoiding domain integrations, suitable for symmetric and nonsymmetric formulations. The method can be applied to arbitrary shaped two-dimensional domains divided into triangular, rectangular and arbitrary shaped quadrilateral linear or curvilinear (e.g. circular) internal cells. The theory is sustained by numerical results for a rectangular and a circular acoustical cavity under Neumann and Dirichlet boundary conditions.