Direct solution vs quadratic programming technique in elastic-plastic finite element analysis

Elastic-plastic plane stress finite element analysis of a disk rolling on a rigid track is performed by the direct method as well as the quadratic programming technique. Tresea and von Mises' yield conditions are used in the former whereas an approximate piecewise linear Tresea yield condition is used in the later case. It is concluded from a comparison of the computer times needed in the two cases that the direct method is far superior to the quadratic programming technique.     

An elastoplastic cracked-beam finite element for structural analysis

A finite element model has been developed in this paper to analyse statically indeterminate skeletal cracked structures. The model is based on elastic-plastic fracture mechanics techniques in order to consider the crack tip plasticity in the analysis. Stiffness matrices for single-edge and double-edge cracked structural elements have been derived using transfer matrix theory. These matrices take into account the effects of axial, flexural and shear deformations due to crack presence. The present model has been applied to investigate the effects of crack size, structure cross-section depth and crack tip plasticity on the redistribution of internal forces in structures. Hence, this analysis can be employed to identify the overstressed regions in cracked structures.     

A general three-dimensional L-section beam finite element for elastoplastic large deformation analysis

In this paper, the development of a general three-dimensional L-section beam finite element for elastoplastic large deformation analysis is presented. We propose the generalized interpolation scheme for the isoparametric formulation of three-dimensional beam finite elements and the numerical procedure is developed for elastoplastic large deformation analysis. The formulation is general and effective for other thin-walled section beam finite elements. To show the validity of the formulation proposed, a 2-node three-dimensional L-section beam finite element is implemented in an analysis code. As numerical examples, we first perform elastic small and large deformation analyses of a cantilever beam structure subjected to various tip loadings, and elastoplastic large deformation analysis of the same structure under reversed cyclic tip loading. We then analyze the failures of simply supported beam structures of different lengths and slenderness ratios under elastoplastic large deformation. The same problems are solved using refined shell finite element models of the structures. The numerical results of the L-section beam finite element developed here are compared with the solutions obtained using shell finite element analyses. We also discuss the numerical solutions in detail.     

Panpenalty finite element programming for plastic limit analysis

A unified panpenalty finite element programming method for limit analysis is established based on the theory of convex analysis, and a penalty-duality finite element model is constructed, which provides an efficient algorithm for the exact solution of the safety factor. In order to reduce the number of degrees-of-freedom in nonlinear programming, a generalized matrix inverse technique is suggested, resulting in a decrease in computer time. Several numerical results for structural analysis are presented.     

A smoothed finite element method for shell analysis

A four-node quadrilateral shell element with smoothed membrane-bending based on Mindlin-Reissner theory is proposed. The element is a combination of a plate bending and membrane element. It is based on mixed interpolation where the bending and membrane stiffness matrices are calculated on the boundaries of the smoothing cells while the shear terms are approximated by independent interpolation functions in natural coordinates. The proposed element is robust, computationally inexpensive and free of locking. Since the integration is done on the element boundaries for the bending and membrane terms, the element is more accurate than the MITC4 element for distorted meshes. This will be demonstrated for several numerical examples.     

A finite element method for localized failure analysis

A method is proposed which aims at enhancing the performance of general classes of elements in problems involving strain localization. The method exploits information concerning the process of localization which is readily available at the element level. A bifurcation analysis is used to determine the geometry of the localized deformation modes. When the onset of localization is detected, suitably defined shape functions are added to the element interpolation which closely reproduce the localized modes. The extra degrees of freedom representing the amplitudes of these modes are eliminated by static condensation. The proposed methodology can be applied to 2-D and 3-D problems involving arbitrary rate-independent material behavior. Numerical examples demonstrate the ability of the method to resolve the geometry of localized failure modes to the highest resolution allowed by the mesh.     

Comparison of boundary element and finite element methods for dynamic analysis of elastoplastic plates

Boundary and finite element methodologies for the determination of the response of inelastic plates are compared and critically discussed. Flexural dynamic plate bending problems are considered and a hardening elastoplastic constitutive model is used to describe material behaviour. The domain/boundary element methodology using linear boundary and quadratic interior elements and the finite element method with quadratic Mindlin plate elements are used in this work. The discretized equations of motion in both methodologies are solved by an efficient step-by-step time integration algorithm. Numerical results obtained are presented and compared in order to access the accuracy and computational efficiency of the two methods. In order to make the comparison as meaningful as possible, boundary and finite element computer codes developed by the author are used in this paper. In general, boundary elements appear to be a better choice than finite elements with respect to computational efficiency for the same level of accuracy.     

The finite volume element method with quadratic basis functions

The paper presents a box scheme with quadratic basis functions for the discretisation of elliptic boundary value problems. The resulting discretisation matrix is non-symmetrical (and also not an M-matrix). The stability analysis is based on an elementwise estimation of the scalar product <A _h u _h ,u _h >. Sufficient conditions placed on the triangles of the triangulation lead to discrete ellipticity. Proof of anO(h ²) error estimate is given for these conditions.     

A Long-step barrier method for convex quadratic programming

In this paper we propose a long-step logarithmic barrier function method for convex quadratic programming with linear equality constraints. After a reduction of the barrier parameter, a series of long steps along projected Newton directions are taken until the iterate is in the vicinity of the center associated with the current value of the barrier parameter. We prove that the total number of iterations isO(nL) orO(nL), depending on how the barrier parameter is updated.On leave from Eötvös University, Budapest and partially supported by OTKA 2116.     

Performance analysis of gradient neural network exploited for online time-varying quadratic minimization and equality-constrained quadratic programming

In this paper, the performance of a gradient neural network (GNN), which was designed intrinsically for solving static problems, is investigated, analyzed and simulated in the situation of time-varying coefficients. It is theoretically proved that the gradient neural network for online solution of time-varying quadratic minimization (QM) and quadratic programming (QP) problems could only approximately approach the time-varying theoretical solution, instead of converging exactly. That is, the steady-state error between the GNN solution and the theoretical solution can not decrease to zero. In order to understand the situation better, the upper bound of such an error is estimated firstly, and then the global exponential convergence rate is investigated for such a GNN when approaching an error bound. Computer-simulation results, including those based on a six-link robot manipulator, further substantiate the performance analysis of the GNN exploited to solve online time-varying QM and QP problems.     

Some reflections on elastoplastic stress calculation in finite element analysis

The set of stress invariants (J₁,J'₂,θ) is very frequently employed in finite element programs for elastoplastic analysis. Some precautions are however required when computing the stress states of contact with the yield surface. The numerical problems that may arise and simple measures to avoid them are described in this paper.     

On the use of a modified Newton method for nonlinear finite element analysis

In this paper, the usefulness of modified Newton methods for solving certain minimization problems arising in nonlinear finite element analysis is investigated. The application considered is nonlinear elasticity, in particular geometrically nonlinear shells. On a test problem, it is demonstrated that a particular implementation of a modified Newton method using both descent directions and directions of negative curvature is able to identify a minimizer, whereas an unmodified Newton method and modified Newton methods using only descent directions fail to converge to the minimizer. The use of modified Newton methods is suggested as a useful complement to the present continuation methods used for nonlinear finite element analysis.     

An implicit and incremental formulation for the solution of elastoplastic problems by the finite element method

In this paper, a new approach is developed to solve elastoplastic problems by the finite element method. This approach involves two steps: (1) A mechanical formulation using the principle of virtual work and an implicit incremental form of the constitutive equations. This form is obtained by an approximate integration of the flow rules over the increment and includes the yield criterion itself. (2) A resolution algorithm to solve the nonlinear equations obtained by the mechanical formulation: Two resolution algorithms based on the Newton-Raphson method are proposed and compared. As the mechanical formulation is no more bound to the resolution algorithm, the results obtained by these two algorithms are the same and are path independent. Two numerical examples are presented: A thick cylinder under an internal pressure and a tensile sample. The numerical results obtained by the presented approach are compared with those obtained by the classical I.S.M. The comparison shows that the accuracy of the results does not vary when the load increment size increases as in the I.S.M. For a given accuracy this method requires about 15 times less computer time than the I.S.M. for the same memory space. This approach is easy to implement in a program based on the I.S.M. and has been extended to compressible and viscoplastic materials.     

The analysis of elastic-plastic plates: A quadratic programming problem and its solution by finite elements

An extended kinematic minimum principle in classical plasticity is used as the basis for the finite element formulation of the rate problem for elastic-plastic plates. A simple algorithm is used to solve the resulting quadratic programming problem. The numerical solution of the problem is carried out in two ways: one method involves load step sizes which are scaled so that one or more gauss points just become plastic and the other method involves load step sizes which are fixed once and for all at the outset. Examples are given and discussed.     

A finite element method for thermoviscoelastic analysis of plane problems

A stress analysis for plane problems in linear thermoviscoelasticity using a finite element formulation is presented. The method employed is based on the assumptions that (1) the material is isotropic, homogeneous and linear, (2) the stress-strain laws are expressed in the hereditary integral form, and (3) the material is thermorheologically simple, which implies that the temperature-time equivalence hypothesis is valid. The associated computer program utilizes isoparametric plane elements.The element matrices that result in the equilibrium equations involve hereditary integrals, and these are approximated by a finite difference scheme for time marching. The solutions for two problems are compared with analytical results evaluated by the integral transform method.For approximate results which require less computer time an alternative form of equilibrium equations utilizing an iterative technique is presented and an example solution is included. Finally, the effect of incompressibility is considered for an axisymmetric numerical example.     

A new method for contour plotting in finite element analysis

A simple method has been developed in order to get a contour plot in an element with given nodal values. The contour lines are obtained by solving a differential equation obtained from the shape function using symbolic computation. This method can be applied to any kind of shape (interpolation) function of a two-dimensional element.     

Design multiple-layer gradient coils using least-squares finite element method

The design of gradient coils for magnetic resonance imaging is an optimization task in which a specified distribution of the magnetic field inside a region of interest is generated by choosing an optimal distribution of a current density geometrically restricted to specified non-intersecting design surfaces, thereby defining the preferred coil conductor shapes. Instead of boundary integral type methods, which are widely used to design coils, this paper proposes an optimization method for designing multiple layer gradient coils based on a finite element discretization. The topology of the gradient coil is expressed by a scalar stream function. The distribution of the magnetic field inside the computational domain is calculated using the least-squares finite element method. The first-order sensitivity of the objective function is calculated using an adjoint equation method. The numerical operations needed, in order to obtain an effective optimization procedure, are discussed in detail. In order to illustrate the benefit of the proposed optimization method, example gradient coils located on multiple surfaces are computed and characterised.     

Interval regression analysis by quadratic programming approach

When we use linear programming in possibilistic regression analysis, some coefficients tend to become crisp because of the characteristic of linear programming. On the other hand, a quadratic programming approach gives more diverse spread coefficients than a linear programming one. Therefore, to overcome the crisp characteristic of linear programming, we propose interval regression analysis based on a quadratic programming approach. Another advantage of adopting a quadratic programming approach is to be able to integrate both the property of central tendency in least squares and the possibilistic property in fuzzy regression. By changing the weights of the quadratic function, we can analyze the given data from different viewpoints. For data with crisp inputs and interval outputs, the possibility and necessity models can be considered. Therefore, the unified quadratic programming approach obtaining the possibility and necessity regression models simultaneously is proposed. Even though there always exist possibility estimation models, the existence of necessity estimation models is not guaranteed if we fail to assume a proper function fitting to the given data as a regression model. Thus, we consider polynomials as regression models since any curve can be represented by the polynomial approximation. Using polynomials, we discuss how to obtain approximation models which fit well to the given data where the measure of fitness is newly defined to gauge the similarity between the possibility and the necessity models. Furthermore, from the obtained possibility and necessity regression models, a trapezoidal fuzzy output can be constructed     

Parallel finite element analysis using Jacobi-conditioned conjugate gradient algorithm

In this paper a modified parallel Jacobi-conditioned conjugate gradient (CG) method is proposed for solving linear elastic finite element system of equations. The conventional element-by-element and diagonally conditioned approaches are discussed with respect to parallel implementation on distributed memory MIMD architectures. The effects of communication overheads on the efficiency of the parallel CG solver are considered and it is shown that for the efficient performance of a parallel CG solver, the interprocessor communication has to be carried out concurrently. A concurrent communication scheme is proposed by relating the semi-bandwidth of the stiffness matrix with the number of independent degrees of freedom and the number of processors and inducing directionalization of communication within the processor pipeline. With the aid of two examples the effectiveness of the proposed method is demonstrated showing that the cost of communication remains low and relatively insensitive to the increase in the number of processors.     

A new strategy for stress analysis using the finite element method

In this paper the authors examine the effectiveness of the Powell-Toint strategy for evaluating the Hessian of the potential energy surface of a finite element model that can be used for linear stress analysis and transient response predictions of structures. Cases for which the Powell-Toint strategy may be cost-effective with the conventional method of stress analysis are identified.