Modelling of muscle behaviour by the finite element method using Hill's three-element model

We present a numerical algorithm for the determination of muscle response by the finite element method. Hill's three-element model is used as a basis for our analysis. The model consists of one linear elastic element, coupled in parallel with one non-linear elastic element, and one non-linear contractile element connected in series. An activation function is defined for the model in order to describe a time-dependent character of the contractile element with respect to stimulation. Complex mechanical response of muscle, accounting for non-linear force–displacement relation and change of geometrical shape, is possible by the finite element method. In an incremental-iterative scheme of calculation of equilibrium configurations of a muscle, the key step is determination of stresses corresponding to a strain increment. We present here the stress calculation for Hill's model which is reduced to the solution of one non-linear equation with respect to the stretch increment of the serial elastic element. The muscle fibers can be arbitrarily oriented in space and we give a corresponding computational procedure of calculation of nodal forces and stiffness of finite elements. The proposed computational scheme is built in our FE package PAK, so that real muscles of complex three-dimensional shapes can be modelled. In numerical examples we illustrate the main characteristic of the developed numerical model and the possibilities of solution of real problems in muscle functioning. © 1998 John Wiley & Sons, Ltd.     

Anisotropic Gurson‐Tvergaard‐Needleman plasticity and damage model for finite element analysis of elastic‐plastic problems

Implementation and analysis of the anisotropic version of the Gurson‐Tvergaard‐Needleman (GTN) isotropic damage criterion are performed on the basis of Hill's quadratic anisotropic yield theory with the definition of an effective anisotropic coefficient to represent the elastic‐plastic behavior of ductile metals. This study aims to analyze the extension of the GTN model suitable for anisotropic porous metals and to investigate the GTN model extension. An anisotropic damage model is implemented using the user material subroutine in ABAQUS/standard finite element code. The implementation is verified and applied to simulate a uniaxial tensile test on a commercially produced aluminum sheet material for three‐dimensional and plane stress test cases. Spherical and ellipsoidal micro voids are considered in the matrix material, and their effects on the uniaxial stress‐strain response of the material are analyzed. Hill's quadratic anisotropic yield theory predicts substantially large damage evolution and a low stress‐strain curve compared with those predicted by the isotropic model. An approximate model for anisotropic materials is proposed to avoid increased damage evolution. In this approximate model, Hill's anisotropic constants are replaced with an effective anisotropy coefficient. All model‐generated stress‐strain predictions are compared with the experimental stress‐strain curve of AA6016‐T4 alloy.     

Geometrically non‐linear thermoelastic analysis of functionally graded shells using finite element method

A finite element formulation governing the geometrically non‐linear thermoelastic behaviour of plates and shells made of functionally graded materials is derived in this paper using the updated Lagrangian approach. Derivation of the formulation is based on rewriting the Green–Lagrange strain as well as the 2nd Piola–Kirchhoff stress as two second‐order functions in terms of a through‐the‐thickness parameter. Material properties are assumed to vary through the thickness according to the commonly used power law distribution of the volume fraction of the constituents. Within a non‐linear finite element analysis framework, the main focus of the paper is the proposal of a formulation to account for non‐linear stress distribution in FG plates and shells, particularly, near the inner and outer surfaces for small and large values of the grading index parameter. The non‐linear heat transfer equation is also solved for thermal distribution through the thickness by the Rayleigh–Ritz method. Advantages of the proposed approach are assessed and comparisons with available solutions are presented. Copyright © 2007 John Wiley & Sons, Ltd.     

Extended finite element method for three‐dimensional crack modelling

An extended finite element method (X‐FEM) for three‐dimensional crack modelling is described. A discontinuous function and the two‐dimensional asymptotic crack‐tip displacement fields are added to the finite element approximation to account for the crack using the notion of partition of unity. This enables the domain to be modelled by finite elements with no explicit meshing of the crack surfaces. Computational geometry issues associated with the representation of the crack and the enrichment of the finite element approximation are discussed. Stress intensity factors (SIFs) for planar three‐dimensional cracks are presented, which are found to be in good agreement with benchmark solutions. Copyright © 2000 John Wiley & Sons, Ltd.     

A finite element model of skeletal muscles

The present paper surveys recent developments in constitutive and computational modelling of skeletal muscles, concerning mainly the generalization to two- and three-dimensional (2D, 3D) continuum deformation analysis of typical one-dimensional (1D) Hill-type muscle models. Extending our previous work in the field and recent contributions by other authors, we describe a constitutive model for skeletal muscles that incorporates all the features of the 3 typical elements (parallel elastic, series elastic and contractile elements) in Hill's muscle model. In particular the proposed incompressible transversely isotropic model incorporates: a multiplicative split of the fibre stretch into contractile and (series) elastic stretches; the possibility of energy storage in the series elastic element; the dependence of the contractile stress on the strain rate; the governing equation of activation dynamics, so that general histories of neural stimulation may be taken as input data. The resulting 2D or 3D constitutive equations are implemented as user subroutines in the large deformation finite element software package ABAQUS. Simple numerical tests are presented and discussed, as well as an example that involves passive or active deformations of a pelvic floor muscle using shell finite elements.     

A low‐order,hexahedral finite element for modelling shells

A thin, eight‐node, tri‐linear displacement, hexahedral finite element is the starting point for the derivation of a constant membrane stress resultant, constant bending stress resultant shell finite element. The derivation begins by introducing a Taylor series expansion for the stress distribution in the isoparametric co‐ordinates of the element. The effect of the Taylor series expansion for the stress distribution is to explicitly identify those strain modes of the element that are conjugate to the mean or average stress and the linear variation in stress. The constant membrane stress resultants are identified with the mean stress components, and the constant bending stress resultants are identified with the linear variation in stress through the thickness along with in‐plane linear variations of selected components of the transverse shear stress. Further, a plane‐stress constitutive assumption is introduced, and an explicit treatment of the finite element's thickness is introduced. A number of elastic simulations show the useful results that can be obtained (tip‐loaded twisted beam, point‐loaded hemisphere, point‐loaded sphere, tip‐loaded Raasch hook, and a beam bent into a ring). All of the gradient/divergence operators are evaluated in closed form providing unequivocal evaluations of membrane and bending strain rates along with the appropriate divergence calculations involving the membrane stress and bending stress resultants. The fact that a hexahedral shell finite element has two distinct surfaces aids sliding interface algorithms when a shell folds back on itself when subjected to large deformations. Published in 2004 by John Wiley & Sons, Ltd.     

A simple boundary element method for problems of potential in non‐homogeneous media

A simple boundary element method for solving potential problems in non‐homogeneous media is presented. A physical parameter (e.g. heat conductivity, permeability, permittivity, resistivity, magnetic permeability) has a spatial distribution that varies with one or more co‐ordinates. For certain classes of material variations the non‐homogeneous problem can be transformed to known homogeneous problems such as those governed by the Laplace, Helmholtz and modified Helmholtz equations. A three‐dimensional Galerkin boundary element method implementation is presented for these cases. However, the present development is not restricted to Galerkin schemes and can be readily extended to other boundary integral methods such as standard collocation. A few test examples are given to verify the proposed formulation. The paper is supplemented by an Appendix, which presents an ABAQUS user‐subroutine for graded finite elements. The results from the finite element simulations are used for comparison with the present boundary element solutions. Copyright © 2004 John Wiley & Sons, Ltd.     

An hp‐adaptive finite element method for electromagnetics. Part 3: A three‐dimensional infinite element for Maxwell's equations

This paper is a continuation of Reference [26] (Cecot, Demkowicz and Rachowicz, Computer Methods in Applied Mechanics and Engineering 2000; 188 : 625–643) and describes an implementation of the infinite element for three‐dimensional, time harmonic Maxwell's equations, proposed in Reference [15] (Demkowicz and Pal, Computer Methods in Applied Mechanics and Engineering 1998; 164 : 77–94). The element is compatible with the hp finite element discretizations for Maxwell's equations in bounded domains reported in References [16–18] (Computer Methods in Applied Mechanics and Engineering 1998; 152 : 103–124, 1999; 169 : 331–344, 2000; 187 : 307–337). Copyright © 2003 John Wiley & Sons, Ltd.     

Finite element crash simulations of the human body: Passive and active muscle modelling

Conventional dummy based testing procedures suffer from known limitations. This report addresses issues in finite element human body models in evaluating pedestrian and occupant crash safety measures. A review of material properties of soft tissues and characterization methods show a scarcity of material properties for characterizing soft tissues in dynamic loading. Experiments imparting impacts to tissues and subsequent inverse finite element mapping to extract material properties are described. The effect of muscle activation due to voluntary and non-voluntary reflexes on injuries has been investigated through finite element modelling.     

A three‐dimensional finite element method with arbitrary polyhedral elements

The ‘variable‐element‐topology finite element method’ (VETFEM) is a finite‐element‐like Galerkin approximation method in which the elements may take arbitrary polyhedral form. A complete development of the VETFEM is given here for both two and three dimensions. A kinematic enhancement of the displacement‐based formulation is also given, which effectively treats the case of near‐incompressibility. Convergence of the method is discussed and then illustrated by way of a 2D problem in elastostatics. Also, the VETFEM's performance is compared to that of the conventional FEM with eight‐node hex elements in a 3D finite‐deformation elastic–plastic problem. The main attraction of the new method is its freedom from the strict rules of construction of conventional finite element meshes, making automatic mesh generation on complex domains a significantly simpler matter. Copyright © 2006 John Wiley & Sons, Ltd.     

A novel three‐dimensional contact finite element based on smooth pressure interpolations

This article proposes a new three‐dimensional contact finite element which employs continuous and weakly coupled pressure interpolations on each of the interacting boundaries. The resulting formulation circumvents the geometric bias of one‐pass methods, as well as the surface locking of traditional two‐pass node‐on‐surface methods. A Lagrange multiplier implementation of the proposed element is validated for frictionless quasi‐static contact by a series of numerical simulations. Published in 2001 by John Wiley & Sons, Ltd.     

有限元简化模型应用于超声换能器模拟分析

依据有限元方法的基本物理思想,在某些不需要计算辐射声场的准确声学参数和波束特性的工程应用方面,对流体模型进行充分简化,提出了简化模型处理的有效方法,利用该方法对超声换能器进行模拟分析,并进行了样品的制作和测试,实测结果与模型简化分析处理的结果基本一致。可以证明,用该方法进行换能器的优化设计是可行和高效的。     

Generalized finite element approaches for analysis of localized thermo‐structural effects

This work addresses computational modeling challenges associated with structures subjected to sharp, local heating, where complex temperature gradients in the materials cause three‐dimensional, localized, intense stress and strain variation. Because of the nature of the applied loadings, multiphysics analysis is necessary to accurately predict thermal and mechanical responses. Moreover, bridging spatial scales between localized heating and global responses of the structure is nontrivial. A large global structural model may be necessary to represent detailed geometry alone, and to capture local effects, the traditional approach of pre‐designing a mesh requires careful manual effort. These issues often lead to cumbersome and expensive global models for this class of problems. To address them, the authors introduce a generalized FEM (GFEM) approach for analyzing three‐dimensional solid, coupled physics problems exhibiting localized heating and corresponding thermomechanical effects. The capabilities of traditional hp‐adaptive FEM or GFEM as well as the GFEM with global–local enrichment functions are extended to one‐way coupled thermo‐structural problems, providing meshing flexibility at local and global scales while remaining competitive with traditional approaches. The methods are demonstrated on several example problems with localized thermal and mechanical solution features, and accuracy and (parallel) computational efficiency relative to traditional direct modeling approaches are discussed. Copyright © 2015 John Wiley & Sons, Ltd.     

On hypersingular surface integrals in the symmetric Galerkin boundary element method: application to heat conduction in exponentially graded materials

A symmetric Galerkin formulation and implementation for heat conduction in a three‐dimensional functionally graded material is presented. The Green's function of the graded problem, in which the thermal conductivity varies exponentially in one co‐ordinate, is used to develop a boundary‐only formulation without any domain discretization. The main task is the evaluation of hypersingular and singular integrals, which is carried out using a direct ‘limit to the boundary’ approach. However, due to complexity of the Green's function for graded materials, the usual direct limit procedures have to be modified, incorporating Taylor expansions to obtain expressions that can be integrated analytically. Several test examples are provided to verify the numerical implementation. The results of test calculations are in good agreement with exact solutions and corresponding finite element method simulations. Copyright © 2004 John Wiley & Sons, Ltd.     

Phase‐field modelling of solidification microstructure formation using the discontinuous finite element method

This paper presents a discontinuous Galerkin finite element computational methodology for solving the coupled phase‐field and heat diffusion equations to predict the microstructure evolution during solidification. The phase‐field modelling of microstructure formation is briefly discussed. The discontinuous Galerkin finite element formulation for the phase‐field model systems for solidification microstructure formation is described in detail. Numerical stability is performed using the Neumann method. The accuracy of the discontinuous model is verified by the analytic solution of a simple 1‐D solidification problem. Numerical calculations using the discontinuous finite element phase‐field model have been performed for simulating the complex 2‐D and 3‐D dendrite structures formed in supercooled melts and the results are compared well with those in literature using the finite difference methods. Parallel computing algorithm is presented and results show that the minimization of the intercommunication between microprocessors is the key to increase the effectiveness in parallel computing with the discontinuous finite element phase‐field model for solidification microstructure formation. Copyright © 2006 John Wiley & Sons, Ltd.     

A mixed hexahedral finite element to model the scattering of a deformed microstrip antenna

Microstrip antennas have a major interest in aeronautical applications due to their low profile. This paper deals with the impact of mechanical strain on the scattering properties of these antennas. Considering a weak coupling between electromagnetism and mechanical behavior, the same 3D hexahedral finite element discretization is used to solve both problems. A node-based approximation is used for mechanical displacement, while for the determination of the electromagnetic fields, a vector finite approximation is implemented to ensure a better consideration of electromagnetic boundary conditions. The weak electromagnetic formulation inducing integrals on open infinite domains, a Boundary Integral Method is used.     

The simple boundary element method for multiple cracks in functionally graded media governed by potential theory: a three‐dimensional Galerkin approach

The simple boundary element method consists of recycling existing codes for homogeneous media to solve problems in non‐homogeneous media while maintaining a purely boundary‐only formulation. Within this scope, this paper presents a ‘simple’ Galerkin boundary element method for multiple cracks in problems governed by potential theory in functionally graded media. Steady‐state heat conduction is investigated for thermal conductivity varying either parabolically, exponentially, or trigonometrically in one or more co‐ordinates. A three‐dimensional implementation which merges the dual boundary integral equation technique with the Galerkin approach is presented. Special emphasis is given to the treatment of crack surfaces and boundary conditions. The test examples simulated with the present method are verified with finite element results using graded finite elements. The numerical examples demonstrate the accuracy and efficiency of the present method especially when multiple interacting cracks are involved. Copyright © 2005 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.