Hyperelastic Problems Research Articles

Strain smoothing for compressible and nearly-incompressible finite elasticity

We present a robust and efficient form of the smoothed finite element method (S-FEM) to simulate hyperelastic bodies with compressible and nearly-incompressible neo-Hookean behaviour. The resulting method is stable, free from volumetric locking and robust on highly distorted meshes. To ensure inf-sup stability of our method we add a cubic bubble function to each element. The weak form for the smoothed hyperelastic problem is derived analogously to that of smoothed linear elastic problem. Smoothed strains and smoothed deformation gradients are evaluated on sub-domains selected by either edge information (edge-based S-FEM, ES-FEM) or nodal information (node-based S-FEM, NS-FEM). Numerical examples are shown that demonstrate the efficiency and reliability of the proposed approach in the nearly-incompressible limit and on highly distorted meshes. We conclude that, strain smoothing is at least as accurate and stable, as the MINI element, for an equivalent problem size.

Inverse Lagrangian formulation for the deformation of hyperelastic solids

Although the field equations of solid mechanics are commonly written in the undeformed configuration, neither fundamental theories nor mathematics prohibits the analysis from being carried out in a deformed configuration. In this letter, following recent developments in inverse deformation problems, the governing equations for static hyperelasticity problems are formulated in the current configuration after deformation. An inverse mapping from the deformed geometry to the original one is solved as the unknown field. Such an approach, herein referred to as the inverse Lagrangian formulation, is exemplified with several applications in addition to the inverse problem of known deformed geometry by design and determination of the original geometry. Applications are found in steady-state fluid–structure interaction problems, such as the design of microfluidic devices. More importantly, the method is also found to be useful in analyzing mechanical instability and bifurcation problems, as the inverse mapping from a buckled state to the original remains unique.

위상 민감도를 이용한 초탄성 비선형 구조의 레벨셋 기반 위상 및 형상 최적설계

A level set based topological shape optimization method for nonlinear structure considering hyper-elastic problems is developed. To relieve significant convergence difficulty in topology optimization of nonlinear structure due to inaccurate tangent stiffness which comes from material penalization of whole domain, explicit boundary for exact tangent stiffness is used by taking advantage of level set function for arbitrary boundary shape. For given arbitrary boundary which is represented by level set function, a Delaunay triangulation scheme is used for current structure discretization instead of using implicit fixed grid. The required velocity field in the actual domain to update the level set equation is determined from the descent direction of Lagrangian derived from optimality conditions. The velocity field outside the actual domain is determined through a velocity extension scheme based on the method suggested by Adalsteinsson and Sethian(1999). The topological derivatives are incorporated into the level set based framework to enable to create holes whenever and wherever necessary during the optimization.

Solving hyperelastic problems using mixed LSFEM

AbstractThe focus of this contribution is the solution of hyperelastic problems using the least‐squares finite element method (LSFEM). In particular a mixed least‐squares finite element formulation is provided and applied on geometrically nonlinear problems. The basis for the element formulation is a div‐grad first‐order system consisting of the equilibrium condition and the constitutive equation both written in a residual form. An L2‐norm is adopted on the residuals leading to a functional depending on displacements and stresses which has to be minimized. Therefore the first variations with respect to both free variables have to be zero. The solution can then be found by applying Newton's Method. For the continuous approximation of the displacements in W1,p with p > 2, standard polynomials are used. Shape functions belonging to a Raviart‐Thomas space are applied for the stress interpolation. These vector‐valued functions ensure a conforming discretization of the Sobolev space H(div, Ω). Finally the proposed formulation is tested in a numerical example. (© 2012 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Implicit Time Integration in the Generalized Interpolation Material Point Method for Finite Deformation Hyperelasticity

In this article, a novel implicit time-integration scheme for the generalized interpolation material point method (GIMP) is discussed. The implicit algorithm discussed in this article is an extension of the algorithm introduced for the material point method (MPM) by Sulsky and Kaul [1]. In this article, the numerical solution employs the Newton-Krylov technique coupled with the conjugate gradient (CG) method to solve the governing equations in a robust, matrix-free fashion. In implicit MPM, the solution procedure needs to be re-ordered in terms of momentum if the algorithm has to solve large deformation problems [1]. This is a direct consequence of the cell-crossing issues in MPM. The solution procedure in this case uses the generalized minimum residual method to solve the resulting non-symmetric equations. However, GIMP is specifically formulated to overcome these particle cell-crossing issues in MPM; hence, a specialized procedure for large deformation is not required. The proposed implicit algorithm for GIMP is benchmarked for hyperelasticity problems by comparison with finite element analysis (FEA) and exact solution.

Mixed spectral/hp element formulation for nonlinear elasticity

We present a mixed formulation for high-order spectral/hp element methods and investigate its stability in large strain analysis of nearly incompressible solids. Specifically, we employ the displacement/pressure formulation and use a pressure projection method along with general Jacobi polynomial bases to derive the discrete equations. We first study the convergence of the method for smooth and singular solutions of elastic and hyperelastic problems for various combinations of mixed elements. Subsequently, we investigate numerically the validity of ellipticity and inf–sup conditions for various test problems including cases with large deformation. We find that the inf–sup condition is satisfied even when the pressure interpolation order Op is equal to the displacement order Ou, but to resolve the locking-free behavior it is required that Op=Ou−1, and this choice also gives the best convergence rate. In large deformation analysis, surprisingly, the results are more stable for Ou⩾Op⩾Ou−k because the ellipticity in kernel is better satisfied, with higher k as Ou increases. Compared to the displacement-only formulation, the mixed formulation seems to have superior accuracy and it is overall more efficient, especially for problems with high bulk modulus.

The Meshless Local Petrov-Galerkin Method for Large Deformation Analysis of Hyperelastic Materials

Nonlinear formulations of the meshless local Petrov-Galerkin method (MLPG) are presented for the large deformation analysis of hyperelastic materials which are considered to be incompressible or nearly incompressible. The MLPG method requires no explicit mesh in computation and therefore avoids mesh distortion difficulties. In this paper, a simple Heaviside test function is chosen for reducing the computational effort by simplifying domain integrals for hyperelasticity problems. Trial functions are constructed using the radial basis function (RBF) coupled with a polynomial basis function. The plane stress hypothesis and a pressure projection method are employed to overcome the incompressibility or nearly incompressibility in the plane stress and plane strain problems, respectively. Effects of the sizes of local subdomain and interpolation domain on the performance of the present MLPG method are investigated. The behaviour of shape parameters of multiquadrics (MQ) function has been studied. Numerical results for several examples show that the present method is effective in dealing with large deformation hyperelastic materials problems.

Parallel Iterative Substructuring in Structural Mechanics

Finite Element Tearing and Interconnecting (FETI) methods are a family of nonoverlapping domain decomposition methods which have been proven to be robust and parallel scalable for a variety of elliptic partial differential equations. Here, an introduction to the classical onelevel FETI methods is given, as well as to the more recent dual-primal FETI methods and some of their variants. With the advent of modern parallel computers with thousands of processors, certain inexact components are needed in these methods to maintain scalability. An introduction to a recent class of inexact dual-primal FETI methods is presented. Scalability results for an elasticity problem using 65 536 processor cores of the JUGENE supercomputer at Forschungszentrum Julich show the potential of these methods. A hyperelastic problem from biomechanics is presented as an application of the methods to nonlinear finite element analysis.

On solving large strain hyperelastic problems with the natural element method

AbstractIn this paper, an extension of the natural element method (NEM) is presented to solve finite deformation problems. Since NEM is a meshless method, its implementation does not require an explicit connectivity definition. Consequently, it is quite adequate to simulate large strain problems with important mesh distortions, reducing the need for remeshing and projection of results (extremely important in three‐dimensional problems). NEM has important advantages over other meshless methods, such as the interpolant character of its shape functions and the ability of exactly reproducing essential boundary conditions along convex boundaries. The α‐NEM extension generalizes this behaviour to non‐convex boundaries. A total Lagrangian formulation has been employed to solve different problems with large strains, considering hyperelastic behaviour. Several examples are presented in two and three dimensions, comparing the results with the ones of the finite element method. NEM performs better showing its important capabilities in this kind of applications. Copyright © 2004 John Wiley & Sons, Ltd.

CAD-Based Shape Optimization Using a Meshfree Method

A shape design sensitivity analysis and optimization procedure is proposed using a meshfree method. A CAD tool connection is established to facilitate the seamless integration of structural modeling, simulation, and design optimization. A new CAD-based method for design velocity computation is proposed, applicable to both the meshfree method and the finite element method. A shape design variable can be chosen among the geometric parameters defined in the CAD model in order to support a concurrent engineering environment. A new CAD-based method for updating the discrete model is proposed, such that the meshfree model is automatically updated in a continuous and consistent way during design optimization. The proposed method is applied to 2D and 3D linear elastic problems, as well as to a large deformation hyperelastic problem.

Higher order stabilized finite element method for hyperelastic finite deformation

This paper presents a higher order stabilized finite element formulation for hyperelastic large deformation problems involving incompressible or nearly incompressible materials. A Lagrangian finite element formulation is presented where mesh dependent terms are added element-wise to enhance the stability of the mixed finite element formulation. A reconstruction method based on local projections is used to compute the higher order derivatives that arise in the stabilization terms, specifically derivatives of the stress tensor. Linearization of the weak form is derived to enable a Newton–Raphson solution procedure of the resulting non-linear equations. Numerical experiments using the stabilization method with equal order shape functions for the displacement and pressure fields in hyperelastic problems show that the stabilized method is effective for some non-linear finite deformation problems. Finally, conclusions are inferred and extensions of this work are discussed.

On the perturbed Lagrangian formulation for nearly incompressible and incompressible hyperelasticity

A perturbed Lagrangian formulation for nearly incompressible and incompressible hyperelasticity is developed by performing a conjugate transformation on the volumetric energy function. The formulation is developed without reference to any specific material models or volumetric energy functions. Solution existence conditions for general nearly incompressible hyperelasticity problems and their discrete counterparts are described. A discussion on the convergence of nearly incompressible solutions to an incompressible solution of continuous as well as discrete hyperelasticity problems is given. The rate of convergence with respect to the compressibility parameter is shown to be one in several numerical examples.

A field boundary element formulation for material nonlinear problems at finite strains

A boundary element formulation for geometrical nonlinear problems is presented, which is applicable for arbitrary constitutive equations based on the concept of hyperelastic response relative to an intermediate configuration. This includes purely hyperelastic problems at large elastic strains. In contrast to other formulations, the Total-Lagrange framework employed in this work significantly increases the numerical efficiency as the system matrices remain constant during the incremental solution and have to be computed only once. A nonlinear set of equations with an identical structure for both boundary and internal unknowns is derived. The basic unknowns are displacement gradients, which are calculated via integral equations and not by differentiation of the shape functions. The formulation can be consistently linearised to allow for the implementation of efficient iteration schemes using gradients, e.g. the Newton-Raphson scheme.

Three-dimensional incompressible viscoelasticity in large strains: Formulation and numerical approximation

This paper considers viscoelastic incompressible solids in large deformations. A three-dimensional thermodynamical model is first proposed. It leads to a nonlinear variational problem which is approximated in space by mixed finite elements and in time by a first order implicit scheme. This approximation is analyzed on the corresponding linearized problem. Finally, an elimination strategy is described which reduces each step of the discrete problem to the solution of a standard hyperelastic problem to be solved by Newton's method.

Alternative methods for the solution of hyperelastic problems with incompressibility

A new approach for analysing incompressible hyperelastic and rubber-like material behaviour is discussed. The approach is based on extending the linear analysis bubble function concept into nonlinear analysis. To eliminate external or residual forces on the bubble function or nodeless degrees of freedom, two iterative methods are discussed. The new approach is compared to other approaches available in the literature including multified principle, reduced integration penalty and average constraint procedures. A unified numerical treatment for handling various constitutive relations of hyperelastic and rubber-like materials is presented. Numerical examples and comparisons include thick-walled cylinder subjected to internal pressure, inflation of a circular rubber plate and inflation of a hollow sphere.

Dual extremum principles and error bounds in nonlinear elasticity theory

Using some concepts of convex analysis a mathematical model is considered to derive dual extremum principles and global error bounds for monotone hyperelastic boundary-value problems. Introducing the two vector spaces of displacement gradients and of Piola stresses, which can be put into duality by a bilinear form, conjugate, and complementary energy functions are defined with the aid of Fenchel's transformation. Sufficient conditions for the convexity of the strain energy density are established. If the strain energy density is a convex function in some convex subset of the vector space of displacement gradients, dual extremum principles can be obtained by using Fenchel's inequality. They provide global error bounds for the solution of hyperelastic boundary-value problems.