Efficient parallel algorithms for elastic–plastic finite element analysis

This paper presents our new development of parallel finite element algorithms for elastic–plastic problems. The proposed method is based on dividing the original structure under consideration into a number of substructures which are treated as isolated finite element models via the interface conditions. Throughout the analysis, each processor stores only the information relevant to its substructure and generates the local stiffness matrix. A parallel substructure oriented preconditioned conjugate gradient method, which is combined with MR smoothing and diagonal storage scheme are employed to solve linear systems of equations. After having obtained the displacements of the problem under consideration, a substepping scheme is used to integrate elastic–plastic stress–strain relations. The procedure outlined controls the error of the computed stress by choosing each substep size automatically according to a prescribed tolerance. The combination of these algorithms shows a good speedup when increasing the number of processors and the effective solution of 3D elastic–plastic problems whose size is much too large for a single workstation becomes possible.     

Application of mixed variational formulations based on the finite element method to the solution of problems on natural vibrations of elastic bodies

We consider mixed variational formulations and the application of the mixed approximations of the finite element method to the solution of problems on natural vibrations of elastic bodies. To solve the generalized spectral problem, three forms of the mixed variational formulations are proposed. The correctness and stability of mixed variational formulations for displacements, strains and stresses are investigated. Matrix equations of the mixed method are given whose solution is performed using the modified algorithm of the steepest descent method. The results of calculations for natural frequencies of free vibrations of a straight and a circular beam are presented that are obtained in the solution of the problem in a two-dimensional formulation based on the classical and mixed finite-element method approaches. __________ Translated from Problemy Prochnosti, No. 2, pp. 121–140, March–April, 2008.     

Application of the moment scheme of the finite element method to the solution of problems with initial stresses in the incremental elasticity theory

We discuss application of the finite element method to the solution of problems with initial stresses within the elasticity theory. Based on the incremental theory of deformable solids, the relationships of the finite element method are derived to calculate the stiffness matrix coefficients for a prestressed spatial element of the serendip family with quadratic approximation of displacements. The calculation of the stressed state of an eccentrically compressed beam and a round plate under conditions of longitudinal-transverse bending is carried out. Comparison of the numerical results with analytical solutions is presented. The variation in the compression and shear strains of a cylindrical damper is studied depending on the degree of deformation and the sequence of load application. __________ Translated from Problemy Prochnosti, No. 3, pp. 131–143, May–June, 2006.     

Interpolation functions in the immersed boundary and finite element methods

In this paper, we review the existing interpolation functions and introduce a finite element interpolation function to be used in the immersed boundary and finite element methods. This straightforward finite element interpolation function for unstructured grids enables us to obtain a sharper interface that yields more accurate interfacial solutions. The solution accuracy is compared with the existing interpolation functions such as the discretized Dirac delta function and the reproducing kernel interpolation function. The finite element shape function is easy to implement and it naturally satisfies the reproducing condition. They are interpolated through only one element layer instead of smearing to several elements. A pressure jump is clearly captured at the fluid–solid interface. Two example problems are studied and results are compared with other numerical methods. A convergence test is thoroughly conducted for the independent fluid and solid meshes in a fluid–structure interaction system. The required mesh size ratio between the fluid and solid domains is obtained.     

Calculation of stress intensity factors based on mode I crack opening integral parameters

We present stress intensity factor assessment using nodal displacements of the crack surfaces determined by the finite element method for cracked bodies. The equation is solved by expanding the crack opening displacement in the Chebyshev function, where crack front asymptotic behavior corresponds to the regulations of the linear elastic fracture mechanics. Results of the stress intensity factor calculations are obtained for test problems with analytical solution. Crack opening displacements are defined with the help of the 3D SPACE software package designed to model mixed variational formulation of the finite element method for displacements and strains of the thermoelastic boundary value problems. Translated from Problemy Prochnosti, No. 6, pp. 122–127, November–December, 2008.     

Mixed-hybrid scheme of the finite element method for solving the problems on bending,free vibration,and stability of plates

To solve a problem on bending, vibration, and stability of plates, a hybrid finite element has been constructed on the basis of Zienkiewicz’s triangle. A mixed approximation is used for the plate deflection and turning angles. It is shown that with a decrease in the triangle dimensions the mixed approach ensures convergence both for the plate deflection and the bending moments, which is practically independent of the way the plate is split into triangular elements. In the problems on free vibrations and stability of plates, the mixed approach yields more exact values of the eigenfrequencies and critical loads as compared to a classical Zienkiewicz’s triangle. The results of the numerical analysis of the convergence and accuracy of the solutions to a number of test problems on bending, free vibration, and stability of a square plate are presented. __________ Translated from Problemy Prochnosti, No. 4, pp. 108–122, July–August, 2008.     

Patch based stress recovery for plate structures

In this paper we address the application of recovery procedures in advanced problems in structural mechanics. The attention is focused on the recovery by compatibility in patches procedure (RCP) and shear deformable plate structures. The formulation of RCP procedure is extended to shear deformable plate problems (Reissner–Mindlin theory) and is applied to recover stresses from mixed and hybrid stress finite elements. These elements offer new possibilities, for recovery procedures in general, which deserve to be discussed. A comprehensive investigation on which finite element solution can be used as input for the recovery procedures is given through standard benchmark problems, obtained for several values of the thickness on structured and unstructured meshes. The numerical results confirm the effectiveness of the recovery procedure extended to plates problems.     

Transformation toughening in the γ-TiAl–β-Ti–V system: Part I Finite element analysis of a crack touching the interface

Atomistic simulation of transformation toughening due to martensitic transformation in Ti–V phase particles dispersed in a γ-TiAl matrix containing cracks requires knowledge of the continuum elastic stress and displacement fields for the problem of a crack touching the γ–β interface. Because of the anisotropic characters of the two phases, analytical solutions for these fields are not available and they must be determined numerically. In the present paper a finite element method-based eigenanalysis is developed and subsequently applied to the γ–β system to determine the order of the stress singularity and the angular dependences of the stress and displacement fields. These fields are subsequently used to enrich the finite elements surrounding the crack tip and, through the use of the general finite element code ABAQUS, to determine the generalized stress intensity factors and thus the total singular crack-tip stress and displacement fields. It is found that there are two coupled singular terms in the singular stress and displacement fields, and consequently pure (uniaxial) mode I loading gives rise to mixed modes I–II near-crack-tip behaviour. This revised version was published online in November 2006 with corrections to the Cover Date.     

Determination of stress intensity factors for semielliptical surface cracks in the WWER-1000 reactor pressure vessel from the results of solving thermoelasticity boundary value problems based on the mixed mesh-projection scheme of the finite element method

Results of numerical analysis of stress intensity factors K_I for semielliptical surface cracks in the WWER-1000 reactor pressure vessel by emergency cooling simulation with known engineering procedures, the equivalent spatial integration and direct methods are presented. Engineering procedures employ the results of numerical solution of axially symmetric boundary value problems of thermoelasticity based on the mixed mesh-projection scheme of the finite element method implemented in the RELAX software. The three-dimensional K_I computations were performed with the SPACE software. __________ Translated from Problemy Prochnosti, No. 2, pp. 45–51, March–April, 2007.     

An alternative version of the Pian–Sumihara element with a simple extension to non-linear problems

 The stiffness matrix for the Pian–Sumihara element can be obtained in a different way than originally presented in Pian and Sumihara (1984). Instead of getting the element matrix from a hybrid stress formulation with five stress terms one can use a modified Hu–Washizu formulation using nine stress and nine strain terms as well as four enhanced strain terms. Using orthogonal stress and strain functions it becomes possible to obtain the stiffness matrix via sparse Bˉ-matrices so that numerical matrix inversions can be omitted. The advantage of using the mixed variational formulation with displacements, stresses, strains, and enhanced strains is that the extension to non-linear problems is easily achieved since the final computer implementation is very similar to an implementation of a displacement element. Received 31 January 2000     

Bordering method for solving systems of linear equations generated by the finite element method in the plate bending problem

A combined iteration algorithm based on the bordering and conjugate gradient methods is proposed to solve systems of linear equations generated by the finite element method in the plate bending problem. The numerical results for the analysis of the convergence rate of the iterative process are presented in the solution of model problems using a classical and modified algorithm of the method of conjugate gradients. The possibility of acceleration of the iterative algorithm is shown. __________ Translated from Problemy Prochnosti, No. 4, pp. 137–145, July–August, 2007.     

On the physical reliability and taking complex loading into account in plasticity

We focus our attention on the physical reliability in the solution of boundary-value problems. We describe a new method for the solution of plasticity problems based on the finite element method with the use of the Il'yushin approximating relation. This enabled us to solve plasticity problems with regard for complex loading and to analyze the physical reliability of the solutions obtained. __________ Translated from Fizyko-Khimichna Mekhanika Materialiv, Vol. 44, No. 4, pp. 43–46, July–August, 2008.     

Mixed Approximation Scheme of the Finite-Element Method for the Solution of Two-Dimensional Problems of the Elasticity Theory

For the solution of two-dimensional boundary-value problems of the elasticity theory, a triangular finite element, ensuring stability and convergence of mixed approximation, is proposed. The system of resolving equations of the mixed method is derived with account of strict satisfaction of static boundary conditions at the surface. To solve matrix equations of the mixed method, various algorithms of the conjugate-gradient method with the pre-conditional matrix have been considered. Numerical data on convergence and accuracy of the solution for a number of test problems of the elasticity theory and fracture mechanics are given. The results obtained by the conventional and mixed finite-element method approaches are compared.     

Micromechanics of fibrous composites subjected to combined shear and thermal loading using a truly meshless method

In this study, a micromechanical model is presented to study the combined normal, shear and thermal loading of unidirectional (UD) fiber reinforced composites. An appropriate truly meshless method based on the integral form of equilibrium equations is also developed. This meshless method formulated for the generalized plane strain assumption and employed for solution of the governing partial differential equations of the problem. The solution domain includes a representative volume element (RVE) consists of a fiber surrounded by corresponding matrix in a square array arrangement. A direct interpolation method is employed to enforce the appropriate periodic boundary conditions for the combined thermal, transverse shear and normal loading. The fully bonded fiber–matrix interface condition is considered and the displacement continuity and traction reciprocity are imposed to the fiber–matrix interface. Predictions show excellent agreement with the available experimental, analytical and finite element studies. Comparison of the CPU time between presented method and the conventional meshless local Petrov–Galerkin (MLPG) shows significant reduction of the computational time. The results of this study also revealed that the presented model could provide highly accurate predictions with relatively small number of nodes and less computational time without the complexity of mesh generation.     

Computer modeling of the jump-like deformation of AMg6 alloy

We describe a procedure for modeling the structural inhomogeneity of a material by the finite element method. We consider the material as a composite consisting of an elastoplastic matrix and brittle inclusions (dispersoids). The finite element model is based on experimental data on the concentration of inclusions and their geometrical sizes. The proposed finite element model describes well the jump-like deformation of AMg6 alloy. __________ Translated from Fizyko-Khimichna Mekhanika Materialiv, Vol. 44, No. 1, pp. 41–44, January–February, 2008.     

A high order hybrid finite element method applied to the solution of electromagnetic wave scattering problems in the time domain

The development of a hybrid high order time domain finite element solution procedure for the simulation of two dimensional problems in computational electromagnetics is considered. The chosen application area is that of electromagnetic scattering. The spatial approximation adopted incorporates both a continuous Galerkin spectral element method and a high order discontinuous Galerkin method. Temporal discretisation is achieved by means of a fourth order Runge–Kutta procedure. An exact analytical solution is employed initially to validate the procedure and the numerical performance is then demonstrated for a number of more challenging examples.     

A numerical verification for an unconditionally stable FEM for elastodynamics

Numerical results for a time-discontinuous Galerkin space–time finite element formulation for second-order hyperbolic partial differential equations are presented. Discontinuities are allowed at finite, but not fixed, time increments. A method for h-adaptive refinement of the space–time mesh is proposed and demonstrated. Numerical results are presented for linear elastic problems in one space dimension. Numerical verification of unconditional stability, as proven in [7], is rendered. Comparison is made with analytic solutions when available. It is shown that the accuracy of the numerical solution can be increased without a major penalty on computational cost by using an adaptively refined mesh. Results are presented for a type of solid–solid dynamic phase transition problem where the trajectory of a moving surface of discontinuity is tracked.     

Frontal solution program for unsymmetric matrices

A frontal solution program is presented which may be used for the solution of unsymmetric matrix equations arising in certain applications of the finite element method to boundary value problems. Based on the Gaussian elimination algorithm, it has advantages over band matrix methods in that core requirements and computation times may be considerably reduced; furthermore numbering of the finite element mesh may be completed in an arbitrary manner. The program is written in FORTRAN and a glossary of terms is provided.     

A Multiscale/stabilized Formulation of the Incompressible Navier–Stokes Equations for Moving Boundary Flows and Fluid–structure Interaction

This paper presents a multiscale/stabilized finite element formulation for the incompressible Navier–Stokes equations written in an Arbitrary Lagrangian–Eulerian (ALE) frame to model flow problems that involve moving and deforming meshes. The new formulation is derived based on the variational multiscale method proposed by Hughes (Comput Methods Appl Mech Eng 127:387–401, 1995) and employed in Masud and Khurram in (Comput Methods Appl Mech Eng 193:1997–2018, 2006); Masud and Khurram in (Comput Methods Appl Mech Eng 195:1750–1777, 2006) to study advection dominated transport phenomena. A significant feature of the formulation is that the structure of the stabilization terms and the definition of the stabilization tensor appear naturally via the solution of the sub-grid scale problem. A mesh moving technique is integrated in this formulation to accommodate the motion and deformation of the computational grid, and to map the moving boundaries in a rational way. Some benchmark problems are shown, and simulations of an elastic beam undergoing large amplitude periodic oscillations in a viscous fluid domain are presented.