Large natural strains and some special difficulties due to non-linearity and incompressibility in finite elements

The finite element method is now a well established tool for the routine treatment of large linear problems, but the treatment of non-linear problems by the method is yet at the beginning.Section 1 of the present work extends the idea of natural strains and stresses to large strains in simplex finite elements.Section 2 applies some algorithms developed for structural dynamics to the problem of non-linear wave propagation including a case of shock development.Section 3 discusses with numerical examples special difficulties when displacement finite elements are used to solve problems with incompressible or nearly incompressible material.     

Time integration in the context of energy control and locking free finite elements

Summary In the present paper two main research areas of computational mechanics, namely the finite element development and the design of time integration algorithms are reviewed and discussed with a special emphasis on their combination. The finite element techniques are designed to prevent locking and the time integration schemes to guarantee numerical stability in non-linear elastodynamics. If classical finite element techniques are used, their combination with time integration schemes allow to avoid any modifications on the element or algorithmic level. It is pointed out, that on the other hand Assumed Stress and Enhanced Assumed Strain elements have to be modified if they are combined with energy conserving or decaying time integration schemes, especially the Energy-Momentum Method in its original and generalized form. The paper focusses on the necessary algorithmic formulation of Enhanced Assumed Strain elements which will be developed by the reformulation of the Generalized Energy-Momentum Method based on a classical one-field functional, the extension to a modifiedHu-Washizu three-field functional including enhanced strains and a suitable time discretization of the additional strain terms. The proposed method is applied to non-linear shell dynamics using a shell element which allows for shear deformation and thickness change, and in which the Enhanced Assumed Strain Concept is introduced to avoid artificial thickness locking. Selected examples illustrate the locking free and numerically stable analysis.     

非线性动态有限元碰撞仿真技术的工程应用研究

该文简要介绍了非线性动态显式有限元碰撞仿真技术的相关理论和算法 ,涉及到实际应用中的一系列关键问题及解决方案。最后文中以薄板冲压、汽车碰撞等工程实际问题为例运用所述理论进行了仿真计算研究 ,取得了令人满意的效果 ,因而也为其它有关碰撞问题的研究解决提供了一种强有力的方法和思路     

A combined dynamic finite element—Riccati transfer matrix method for solving non-linear eigenproblems of vibrations

The non-linear eigenvalue problems arising from dynamic finite elements in structural analysis are considered. A method of combining the dynamic finite element method and the Riccati transfer matrix method is proposed to solve non-linear eigenproblems of vibrations. This method finds the value of the frequency using the Newton-Raphson method. By this technique, the number of nodes required in the regular dynamic finite element method is reduced, and therefore a microcomputer may be used. Besides, no plotting of the value of the determinant versus the assumed frequency is necessary. The Riccati transformation of state vectors is proposed in order to avoid the propagation of round-off errors occurring in recursive multiplications of the transfer matrices. A program DFERTM-W based on this method for use on an IBM PC-AT microcomputer is developed. Finally, numerical examples are presented to demonstrate the accuracy, as well as the capability, of the proposed method for solving non-linear eigenproblems of vibrations.     

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.     

WICFEM-a FORTRAN program for solutions of Saint-Venant streamflow equations

A FORTRAN program based on the coupled algorithms of the Galerkin finite element method and the Newton-Raphson iterative technique for simultaneous solutions of two one-dimensional, non-linear, hyperbolic, partial differential equations for streamflow system is described. A combination of linear basis function and forward differencing for time derivatives are incorporated in the program. Conditions for attaining unconditional stability and fast convergence rate for prescibed integration time are explained. The paper includes methods of modifying geometric parameters, since no two river channel geometries are alike and typical application to river flood forecasting.     

Non-linear finite element analysis of composite beams with interlayer slips

This article presents a new non-linear finite element formulation for the analysis of two-layer composite plane beams with interlayer slips. The element is based on the corotational method. The main interest of this approach is that different linear elements can be automatically transformed to non-linear ones. To avoid curvature locking that may occur for low order element(s), a local linear formulation based on the exact stiffness matrix is used. Five numerical applications are presented in order to assess the performance of the formulation.     

A parallel finite element solution method

New parallel computer architectures have revolutionized the design of computer algorithms, and promise to have significant influence on algorithms for structural engineering computations. In this paper, a parallel finite element solution method is presented. The solution method proposed does not require the formation of global system equations, but computes directly the element distortions, as opposed to solving a system of nodal equations. An element or substructure is mapped on to a processor of an MIMD multiprocessing system. Each processor stores only the information relevant to the element or substructure for which the processor represents. The finite element computations can be performed in parallel, in that a processor generates the local stiffness, computes the element distortions and determines the stress-strain characteristics for the element or substructure associated with the processor.     

Numerical methods for calculating pressure distribution in gas bearings

In gas bearings, the pressure distribution is governed by a non-linear Reynolds equation. In order to solve this equation two numerical methods, the conservative difference scheme and the finite element method, are provided in this paper. They are superior to the finite difference method of Colemman [2]. Use of the finite element method is advocated because of its flexibility in solving the Reynolds equation.     

Fundamentals of the finite element method

The present paper gives a brief review of the finite element method. After a historical review, the organization of the finite element analysis in two steps, an element analysis and a system analysis, is described for a simple frame problem.The generalization of this idea to two-and three-dimensional problems is explained and the application of simple types of elements is discussed. The extension to more complex elements is outlined, and finally the capability of the method in solving non-linear and dynamic problems is sketched.     

Application of a modified semismooth Newton method to some elasto-plastic problems

Some elasto-plasticity models with hardening are discussed and some incremental finite element methods with different time discretisation schemes are considered. The corresponding one-time-step problems lead to variational equations with various non-linear operators. Common properties of the non-linear operators are derived and consequently a general problem is formulated. The problem can be solved by Newton-like methods. First, the semismooth Newton method is analysed. The local superlinear convergence is proved in dependence on the finite element discretisation parameter. Then it is introduced a modified semismooth Newton method which contain suitable “damping” in each Newton iteration in addition. The determination of the damping coefficients uses the fact that the investigated problem can be formulated as a minimisation one. The method is globally convergent, independently on the discretisation parameter. Moreover the local superlinear convergence also holds. The influence of inexact inner solvers is also discussed. The method is illustrated on a numerical example.     

Implementation of sub-structuring within an object-oriented framework

This paper describes how the U^tDU decomposition method and sub-structuring algorithms can be implemented using object-oriented techniques. It is shown that this enables the algorithms to be implemented very concisely. Moreover, there is no increase in code complexity when the algorithms are extended to take account of sparsity. The sub-structuring, or domain decomposition algorithms are expressed in block matrix terms, and classes are used to represent each of these matrices. The solution processes are incorporated in a finite element program. The finite element program uses a distributed data structure, and this facilitates a straightforward interface between the finite element program and the mathematics. Moreover, the program possesses a clear control structure for responding to user changes to the finite element model.     

Finite element/difference methods in random vibration

Finite element and finite difference analyses of a class of random vibration problem are presented. The random field equations are discretized using the standard finite element/difference techniques. Recursive algorithms for the moments of the response in terms of the moments of the random loading and random initial condition are derived based on the discretized equations. The approach is straightforward and it provides an alternative to the standard normal-mode approach in the analysis of linear random structural systems. However, unlike the standard approach, the finite element/difference approach can be applied to other more complex systems. Two numerical examples are given to illustrate the application. The approximate numerical results are also compared with the exact solutions to check the accuracy of the finite element/difference analyses. It is found that for the problems considered, the finite difference algorithms provide better accuracy and efficiency than the finite element method.     

Volume preserving mesh parameterization based on optimal mass transportation

In order to convert a finite element mesh model to the spline representation for the purpose of isogeometric analysis, one needs to parameterize the solid. This work introduces a novel volumetric parameterization method, which guarantees to be free of volume distortion.Given a simply connected tetrahedral mesh with a single boundary surface, we first compute a harmonic map from the boundary triangle mesh to the unit sphere by non-linear heat diffusion method; then we use the surface harmonic map as the boundary condition to compute the volumetric harmonic map to parameterize the solid onto the unit solid ball; finally we compute an optimal mass transportation map from the unit solid ball with the push-forward volume element induced by the harmonic map onto itself with the Euclidean volume element. The composition of the volumetric harmonic map and the optimal mass transportation map gives an volume-preserving parameterization.The method has solid theoretic foundation, and is based on conventional algorithms in computational geometry, easy to implement. We have thoroughly tested our algorithm on many solid models in reality. The experimental results demonstrate the efficiency and efficacy of the proposed method. To the best of our knowledge, it is the first work addressing volume-preserving parameterization in the literature.     

Structural dynamic analysis on a parallel computer: The finite element machine

The development of general-purpose finite element computer software systems has provided the capability to analyze a wide range of linear and non-linear structural problems. However, these software systems are severely limited for non-linear response calculations because of the available speed on current sequential computers. Recent and projected advances in parallel multiple instruction multiple data (MIMD) computers provide an opportunity for significant gains in computing speed and for broadening the range of structural problems which may be solved. The key to these gains is the effective selection and implementation of algorithms which exploit parallel computing. This paper documents experiences solving transient response calculations on an experimental MIMD computer, termed the Finite Element Machine. The paper describes the algorithm used, its implementation for parallel computations, and results for representative one- and two-dimensional dynamic response test problems. The results show computation speedups of up to 7.83 for eight processors, and indicate that significant speedups of solution time are possible for non-linear dynamic response calculations through the use of many processors and appropriate parallel integration algorithms. The results are extremely encouraging and suggest that significant speedups in structural computations can be achieved through advances in parallel computers.     

A Cosserat non-linear finite element analysis software for blocky structures

The non-linear analysis of bidimensional finite element models of blocky structures by using the Cosserat theory is described in this paper. The Cosserat theory is employed to describe the behaviour of blocky structures, showing that this finite element analysis is suitable to determine the response of 2D models of elastoplastic problems including nodal rotations and displacements. The elastoplastic analysis of such models is carried out by using a tilting-block criterion derived from classical analysis. Plastic surfaces as well as flow vectors are formulated in order to analyse blocky structures currently used in masonry engineering situations, discussing how input parameters affect the overall behaviour of these models. The software developed is also presented and discussed, showing that the algorithms proposed are robust and efficient. Finally, some numerical examples are presented and discussed in detail.     

Sensitivity of structural response in context of linear and non-linear buckling analysis with solid shell finite elements

The paper is concerned with the sensitivity analysis of structural responses in context of linear and non-linear stability phenomena like buckling and snapping. The structural analysis covering these stability phenomena is summarised. Design sensitivity information for a solid shell finite element is derived. The mixed formulation is based on the Hu-Washizu variational functional. Geometrical non-linearities are taken into account with linear elastic material behaviour. Sensitivities are derived analytically for responses of linear and non-linear buckling analysis with discrete finite element matrices. Numerical examples demonstrate the shape optimisation maximising the smallest eigenvalue of the linear buckling analysis and the directly computed critical load scales at bifurcation and limit points of non-linear buckling analysis, respectively. Analytically derived gradients are verified using the finite difference approach.     

A collocation solution for Burgers' equation using cubic B-spline finite elements

The non-linear Burgers' equation is solved numerically by a B-spline finite element method. The approach used is based on collocation of cubic B-splines over finite elements so that we have continuity of the dependent variable and its first two derivatives throughout the solution range. A linear stability analysis shows that a numerical scheme based on a Crank-Nicolson approximation in time is unconditionally stable. Three standard problems are used to validate the algorithm. Comparisons are made with published numeric and analytic solutions. The proposed method performs well.     

Explicit multi-time step integration for the non-linear thermal analysis of structures

An explicit integration algorithm is presented which uses different time steps to integrate different element subdomains of a mesh. The method is applicable to non-linear first order finite element systems. Stability is shown by considering an energy norm which decreases and the critical time step is given in terms of element eigenvalues. Several numerical examples are presented.     

Automating the Finite Element Method

The finite element method can be viewed as a machine that automates the discretization of differential equations, taking as input a variational problem, a finite element and a mesh, and producing as output a system of discrete equations. However, the generality of the framework provided by the finite element method is seldom reflected in implementations (realizations), which are often specialized and can handle only a small set of variational problems and finite elements (but are typically parametrized over the choice of mesh). This paper reviews ongoing research in the direction of a complete automation of the finite element method. In particular, this work discusses algorithms for the efficient and automatic computation of a system of discrete equations from a given variational problem, finite element and mesh. It is demonstrated that by automatically generating and compiling efficient low-level code, it is possible to parametrize a finite element code over variational problem and finite element in addition to the mesh.