Michał Kleiber

Perturbation based stochastic finite element method for homogenization of two-phase elastic composites

Abstract The main idea of the paper is to apply the second order perturbation and stochastic second central moment technique to solve the homogenization problem. In order to determine the effective elasticity tensor, the prevailing computational methodology discussed in the literature so far was the Monte-Carlo simulation providing appropriate expected values and higher order probabilistic moments of the effective tensor components. The technique applied in this paper aims at significantly reducing the computational cost of the simulation without sacrificing the solution accuracy. The numerical example substantiates this claim in the case of a periodic fiber-reinforced plane strain composite with random fiber and matrix Young’s moduli.

Stochastic finite element modelling in linear transient heat transfer

A stochastic heat transfer variational principle is suggested for transient and steady-state heat problems. The non-statistical formulation is based on the combination of the second-order perturbation technique and second-moment analysis. The principle allows incorporation of system uncertainties into the conventional finite element equations, which are solved for the first two probabilistic moments of the nodal random temperature field. Computational aspects of the problem are discussed. Numerical algorithms have been worked out and easily adapted to an existing finite element code. The formalism presented may be directly employed for a wide class of field problems.

Reliability assessment for sheet metal forming operations

Methodology developed for reliability calculations of structures is applied to estimate reliability of sheet metal forming operations. Sheet forming operations are one of the most common technological processes but still the tool and process design is a difficult engineering problem. Product defects are often encountered in the industrial practice. Material breakage, wrinkling, shape defects due to springback are most frequent defects in sheet metal forming operations. Numerical simulation allows us to evaluate product manufacturability and predict the defects at early stages of the design process. In the paper the so-called forming limit diagrams (FLD) are used as a criterion of material breakage in the manufacturing process. A zone of a FLD where good results are guaranteed with sufficient probability is considered as safe zone. Sheet forming operations are characterized with a significant scatter of the results. This can be caused by differences that can occur in forming of each part. Small differences in the contact conditions, for instance, can lead to significant changes in the deformation state of the sheet. In reliability-like approach we try to quantify intuitive terms of probability of failure/success of forming operations given some uncertainty of parameters characterizing a forming process like friction parameters or blankholding force. Since the employment of the gradient-based reliability techniques is very much limited due to the some degree of numerical noise introduced by the explicit dynamic algorithm used to perform sheet stamping simulation the method of adaptive Monte Carlo simulations were chosen for reliability assessment.

ELASTO-PLASTIC INTERFACE MODEL FOR 3D-FRICTIONAL ORTHOTROPIC CONTACT PROBLEMS

The present study deals with the solution of the fully three-dimensional contact/friction problem taking into account microstructural characteristics of the surfaces. An incremental non-associated hardening friction law model analogous to the classical theory of plasticity is used. Two different non-linear friction functions in the orthogonal directions are used to account for the orthotropic properties of the contacting bodies. A frontal solver processing unsymmetric matrices is adopted. Two numerical examples have been selected to show applicability of the method proposed.

Sensitivity analysis in plane stress elasto-plasticity and elasto-viscoplasticity

Abstract Development of computational models for plane stress elasto-plasticity is known to make some difficulties non-existent in 3D and plane strain formulations. Tedious derivation of plane stress algorithmic tangent matrices which cannot be directly obtained as a special case of 3D matrices may serve here as an example. In this paper specific features of plane stress computations are discussed in the context of parameter sensitivity analysis. Elasto-plastic material model with isotropic/kinematic hardening as well as two viscoplastic models are considered. Explicit expressions for sensitivity gradients are derived. Instructive examples solved by FEM illustrate the paper.

Stochastic structural interface defects in fiber composites

Abstract In this paper the new idea of homogenization of stochastic interface structural defects in the matrix of a fiber composite, recently proposed by the first author, is extended to the contact zones of both components. The physical mechanism of the occurrence of interface defects is described and different versions of location of these defects on fiber-matrix boundary are tested using the stochastic finite element method (SFEM). Two first moments of the displacement random fields are computed and a new SFEM formulation for such problems is proposed.

Sensitivity of forming processes to shape parameters

Abstract Shape sensitivity analysis is presented for metal-forming processes described in terms of the flow approach [1]. The direct differentiation method is employed to derive the sensitivity expressions. The continuum approach is adopted so that both the equilibrium equations and response functional are differentiated before discretization. The necessary derivatives with respect to shape design variables are calculated making use of the framework already available for isoparametric elements (control volume approach). In finite element implementation a system of equations is obtained which has the same system matrix as the equilibrium problem. The right-hand side is calculated by partially differentiating the internal and external forces with respect to the design parameters. The procedure is illustrated by calculating sensitivities of some independent and dependent variables with respect to the die angle in an extrusion problem and to the roll radius in a plane rolling.

Finite element analysis based on stochastic Hamilton variational principle

Abstract A stochastic Hamilton variational principle (SHVP) is formulated for dynamic problems of linear continuum. The SHVP allows incorporation of probabilistic distributions into the finite element analysis. The formulation is simplified by transformation of correlated random variables to a set of uncorrelated random variables through a standard eigenproblem. A procedure based on the Fourier analysis and synthesis is presented for eliminating secularities from the perturbation approach. The algorithms developed can readily be adapted to existing deterministic finite element codes.

Sensitivity analysis of metal forming processes involving frictional contact in steady state

Abstract A simple element to model frictional contact in steady state metal forming processes is presented together with the sensitivity analysis to the friction coefficient in a Coulomb friction law. The interest of such model arises from the analysis of rolling processes and a two dimensional approach to cutting problems, where the contact zone is to be determined, however a stationary state is present in most part of the operation. The flow approach proves to be an adequate method to handle efficiently this situation. The contact elements impose a restriction in the velocity component normal to the boundary and a tangential friction force opposite to the velocity. The parts of the boundary which are not in closed contact are treated as free surfaces, which must fulfill the condition of being streamlines. Sensitivity analysis with respect to the friction coefficient is performed by the Direct Differentiation Method (DDM). The effect of variations in this parameter is discussed for the simulation of an extrusion and a cutting problem.

On a numerical approach to shakedown analysis of structures

Abstract Two alternative ways of performing the shakedown analysis of structures subjected to variable repeated loads are presented. The first one enables one to check whether the structure shakes down (i.e. responds elastically after a few elastoplastic cycles) or not. Such a check is done by reproducing incrementally a critical cyclic load which corresponds to a cyclic repetition of piecewise-proportional load path that contains all the vertices of the variable load domain. The second approach enables one to find a safety factor for the limit of shakedown capability. The problem is one of convex or linear programming, depending on the kind of yield condition used. Numerical results presented in the paper show that the general purpose software that performs incremental elastoplastic calculations can be successfully used for shakedown analysis.