Narrow Results

This paper presents the theory and applications of the Hellinger-Reissner mixed formulation for the nonlinear frame element with lateral deformable supports. The governing differential equations of the problem (strong form) are derived first. Then, the Hellinger-Reissner mixed frame element (weak form) is formulated to solve for the numerical solution of the problem. Tonti' s diagrams are employed to conveniently represent the equations governing both strong and weak forms of the problem. Finally, a numerical example is used to show that the Hellinger-Reissner mixed element is much more accurate than the classical displacement-based element. The nonlinear frame model proposed in this paper has practical applications in modelling the soil-pile structural system, geosynthetics/fiber-glass reinforcement of foundation soils, beam on deformable foundations, etc.

We present a finite element family for the Kirchhoff problem by using C⁰-continuous basis functions for the deflection and the rotation, i.e., the same approach as typically used for the Reissner–Mindlin model. A theoretical error analysis shows that optimal a-priori error estimates are valid for the method, in the presence of free boundary conditions as well. Furthermore, a reliable and efficient a-posteriori error estimator is introduced for adaptive mesh refinements. Benchmark computations verify the optimal rate of convergence and illustrate the robustness of the a-posteriori error estimator.

In this work, a model for the quasistatic frictionless contact between a viscoelastic body with long memory and a foundation is studied. The material constitutive relation is assumed to be nonlinear and the contact is modelled with the normal compliance condition, i.e., the obstacle is assumed to be deformable. The evolution of the mechanical damage of the material, caused by the opening and growth of microcracks under excessive stress or strain, is modelled by a nonlinear partial differential equation. We discuss the existence and uniqueness of weak solution for the mechanical problem and we introduce a fully discrete scheme based on the finite element method and finite differences to approximate this solution. The existence and uniqueness of the approximate solutions are presented and some estimates of the numerical error are provided. Finally, we show some results obtained in the simulation of a two-dimensional test problem.

This paper presents a parallel solver for the Monodomain and Bidomain systems modeling the bioelectrical activity of cardiac tissue in three dimensions. These systems consists of nonlinear parabolic reaction-diffusion equations coupled with a stiff system of several ordinary differential equations, known as Luo-Rudy I model. Instead of the more common explicit or semi-implicit time discretizations, we consider here the fully implicit Backward Euler and linearly implicit Rosenbrock methods, that allow a time step-size selection without stability constraints. Trilinear finite elements are employed in the space discretization. The nonlinear systems originated at each time step of Backward Euler are solved by a Newton-Krylov method, where the outer iteration of an inexact Newton method solves the linear systems of the Jacobian update with a Krylov method such as GMRES with a block Jacobi preconditioner. The same linear solver is applied to the linear systems of the Rosenbrock method. Parallel numerical results on a Linux Cluster show that the proposed parallel solver is scalable for the Monodomain system, with both linear and nonlinear iteration counts bounded from above by a constant independent of the number of processors. This does not hold for the Bidomain system, since the linear iteration counts increase with the number of processors.

Based on the ideas from Refs. 1 and 2 we present a new finite element method for the Kirchhoff plate bending model.³ The method uses C⁰ basis functions for the deflection and the rotation, i.e. the same approach as used for the Reissner–Mindlin model. To account for the effective shear force at the free boundary a stabilization term is added. We prove optimal a-priori and a-posteriori error estimates. The corresponding results from benchmark problems are also reported.

Biomechanical modeling of human and animal brain tissue is a growing field of research, whose applications currently include simulating, with a view to minimizing, head injuries in car impacts, generically modeling dynamic behavior in the surgical theater, such as brain shift, and increasingly, providing medical experts with clinical tools such as surgical simulators and predictive models for tumour growth. This chapter provides an overview of the literature on the biomechanics of the brain, with a particular emphasis on applications to intrasurgical brain shift estimation and to surgical simulation. Included is a discussion of the underlying continua, of numerical estimation techniques, and of related cutting and resection models.

In this paper we present a discrete stochastic model for laser-induced jet-chemical etching in one and two spatial dimensions for the microstructuring of metals. Experimental observations show that depending upon the choice of system parameters either surface roughening occurs or the formation of periodic ripple structures. The results of kinetic Monte-Carlo simulations for the discrete model are compared to experimental observations. The continuum limit taken for the discrete model leads to a modified Kuramoto-Sivashinsky (KS) equation describing the evolution of the etching front. This type of equation is also characteristic for other erosion processes like water-jet cutting and can be considered as an order parameter equation. The spontaneous ripple formation can then be explained by the development of a front instability. Finite element simulations of the modified KS equation support the theoretical findings.

This paper describes an innovative computational model developed to solve two-dimensional incompressible viscous flow problems in external flow fields. The model based on the Navier-Stokes equations in primitive variables is able to solve the infinite boundary value problems by extracting the boundary effects on a specified finite computational domain, using the pressure projection method. The external flow field is simulated using the boundary element method by solving a pressure Poisson equation that assumes the pressure as zero at the infinite boundary. The momentum equation of the flow motion is solved using the three-step finite element method. The arbitrary Lagrangian-Eulerian (ALE) method is incorporated into the model, to solve the moving boundary problems. For illustration of the present numerical code, a vortex-induced cross-flow oscillations of a circular cylinder mounted on an elastic dashpot-spring system is considered. The phenomena of the beat, lock-in, and resonance are revealed in the Reynolds number range between 100 and 110, which are much narrower than the previous results by experimental and numerical studies.

The interactions between a macro-crack and a cluster of micro-defects are studied numerically by using a series of special finite elements each containing a defect. These special finite elements, which contain defects such as holes, cracks, and inhomogeneities, are developed based on the hybrid displacement, complex potential and conformal mapping techniques. These hybrid-type elements can be used together with the conventional finite elements without any difficulty. In this paper, the mathematical and finite element modeling procedures for the study of the above-mentioned problems are presented. The numerical results obtained are compared with the existing theoretical results.

This paper presents a computational approach for composite materials subjected to ballistic impact. The composite materials are composed of layers of fabric embedded in a matrix material. The fabric is represented by bar elements and the matrix material is represented by solid elements. The geometric characteristics of these composite materials are three dimensional, which in turn requires a three dimensional analysis. The strength and failure characteristics of the fabric are determined directly from the actual fiber characteristics. For the matrix material it is necessary to modify the solid elements, because much of the volume of the solid elements is occupied by the volume of the fabric. The resulting modifications include the determination of effective density, stiffness, strength and failure characteristics for the solid elements. This approach has been shown to provide good agreement with experimental ballistics data. Because this approach treats each of the component materials individually (instead of blending them into a homogeneous anisotropic material), it is possible to examine the effects of individual material properties for both the fabric and the matrix material. This includes the capability to allow for failure of the fabric material and/or the matrix material. Parametric computational results are included to illustrate some of these effects.

A novel model for 3-phase fluid flow through a deformable fissured porous media is presented. The set of governing equations, in addition to the traditionally used continuity equation, consists of equilibrium equation and therefore a coupled model is utilized. The well-known Galerkin finite element method for spatial discretization along with Kantrovich-type temporal discretization is applied where solid phase deformation as well as fluids' pressure are considered as the primary unknowns. Other unknowns are calculated based on the primary unknowns, hence a highly non-linear system is introduced. The obtained results show a significantly different behavior for the porous media where the fluid flow is coupled with deformability of the media.

A second order finite element method for the instationary Navier-Stokes equation with free capillary surface is proposed and applied to simulate the flow dynamics of a droplet after impinging on a solid surface.

A simple non-linear mechanical system comprising a pin-jointed string of finite-length links, supported by elastic springs at the pins and compressed by an axial load, is viewed from two perspectives. When seen as an initial-value problem, equilibrium equations provide an iterative non-linear mapping. When seen as a boundary-value problem, it becomes a simple finite element model. At loads less than the critical buckling load, a preferred buckling configuration is found that is localized along the length. In the limit of infinite length this is described as a homoclinic connection in phase space, joining the flat equilibrium state to itself. The infinite sequence of homoclinic points thus defined embeds within the complex topological structure of a homoclinic tangle, within which also appear periodic, quasi-periodic, and chaotic spatial solutions. Implications in the finite element setting are discussed. ©1997 by John Wiley & Sons, Ltd.

The thermally-induced deformation of flexible space structures could be suppressed by controlling the temperature difference on certain cross-sections of these kinds of thinwalled structures. For this purpose, one must optimize the applied heat flux to control the temperature difference and deformation of thin-walled structures. This optimize procedure could be very time consuming, because the radiation analysis has to be conducted in each iteration step. However, this problem can be greatly remedied with the help of the Fourier temperature finite element, which has been proposed in our early work. Based on this element, this paper develops a heat flux optimize method to control the thermally-induced deformation of thin-walled structures. A numerical example demonstrates the validity and efficiency of the proposed method.

Fourier temperature finite element is very efficient on nonlinear analysis of transient temperature fields of thin-walled tubes absorbing space heat flux and emitting thermal energy by radiation. Controlled temperature of thin-walled tubes can supress the thermally-induced responses of space structures. Based on fourier-temperature finite element, this paper develops a heat flux identification method for given temperature on thin-walled tubes and the identification result can be used to actively control the temperature of thin-walled tubes. A numerical example illustrate that the heat flux value, acting position and acting area can be reliably identified by this method.

In this paper we are mainly concerned with the additive overlapping domain decomposition algorithms and their implementation on parallel computers with shared and distributed memory including clusters of workstations. The problem under consideration is the numerical solution of 3D elasticity systems but the main ideas, algorithms and codes are common for the 3D elliptic boundary value problems. The system of linear equations which has to be solved after a finite element method discretization is symmetric and positive definite and the Preconditioned Conjugate Gradient method with a preconditioner constructed via Domain Decomposition is used. The algorithm is highly parallelizable. The Message Passing Interface (MPI) standard is used on both shared and distributed memory parallel computer platforms. Results from numerical experiments on two symmetric multiprocessor systems and IBM SP parallel computer with distributed memory are discussed.

The paper surveys recent results on the finite element approximation of Hamilton-Jacobi-Bellman equations. Various methods are analyzed and error estimates in the maximum norm are derived. Also, a finite element monotone iterative scheme for the computation of the approximate solution is given and its geometrical convergence proved.