Jiří Plešek

Isogeometric analysis of free vibration of simple shaped elastic samples

The paper is devoted to numerical solution of free vibration problems for elastic bodies of canonical shapes by means of a spline based finite element method (FEM), called Isogeometric Analysis (IGA). It has an advantage that the geometry is described exactly and the approximation of unknown quantities is smooth due to higher-order continuous shape functions. IGA exhibits very convenient convergence rates and small frequency errors for higher frequency spectrum. In this paper, the IGA strategy is used in computation of eigen-frequencies of a block and cylinder as benchmark tests. Results are compared with the standard FEM, the Rayleigh-Ritz method, and available experimental data. The main attention is paid to the comparison of convergence rate, accuracy, and time-consumption of IGA against FEM and also to show a spline order and parameterization effects. In addition, the potential of IGA in Resonant Ultrasound Spectroscopy measurements of elastic properties of general anisotropy solids is discussed.

Assessments of methods for locating the point of initial yield

Numerical methods for locating the intersection point of the loading path with the temperature sensitive yield surface are discussed. The Newton method is found to be suitable for this purpose. It is concluded that the iterates should begin at the plastic state rather than at the stress point estimated by a linear interpolation. The influence of temperature changes on the iterative process is analyzed.

Explicit integration method with time step control for viscoplasticity and creep

Constitutive equations for viscoplasticity and creep are expressed in terms of stress components, temperature and internal variables. The resulting set of ordinary differential equations of the first order under the guise of the finite element method is integrated by the Euler forward scheme with automatic subincrementation. The time step length is set on the basis of a posteriori error estimate. Testing examples were computed, including benchmark problems introduced by Zienkiewicz and Cormeau. In every case a perfect match between the converged step length and the Cormeau formula for the critical time step was observed.

Numerical Implementation of A Model With Directional Distortional Hardening

AbstractThe presented work outlines and tests a finite-element (FE) implementation of a simple directional distortional hardening model. The evolution equations of internal variables are based on the Armstrong–Frederick evanescent memory-type hardening rule, and the associative flow rule is adopted. The directional distortion of the yield function is governed by the contraction of the backstress tensor with the so-called unit radial tensor, and incorporates a fixed scalar distortional parameter. Therefore, the variety of possible shapes of predicted subsequent yield surfaces is limited. The tangent stiffness–radial corrector method with fine subincrementation is used. This implementation is verified by comparing numerical results with analytical solutions pertinent to proportional load cases and semianalytical solutions for nonproportional loadings. The template of the whole implementation is included in the Appendix.

Finite Element Computational Technology in Resonant Ultrasound Spectroscopy of Composite Materials

RUS−resonant ultrasound spectroscopy is a recent experimental−numerical method for the determination of moduli of elastic materials. Generally, all 21 elastic components of the elastic tensor can be determined by the numerical procedure based on the knowledge of a mechanical spectrum of a specimen. This involves the solution of a demanding inverse problem. Currently, the RUS technology allows the determination of material parameters of composite materials consisting of several layers with different material properties. In the present work, the fixed point iteration method in connection with the finite element method developed earlier is extended to optimize elastic moduli of layered materials. Properties of the fixed point iteration method are tested on a bicrystal specimen.

Sparse direct solver for large finite element problems based on the minimum degree algorithm

Abstract A sparse direct solver for large problems from solid continuum mechanics based on the minimum degree algorithm is proposed and tested. The solver is designed to take advantage of the properties of the finite element method, particularly the structure of the finite element mesh. For the minimization of the fill-in in the matrix factors a modification of the approximate minimum degree ordering algorithm of Amestoy, Davis and Duff is utilized. The employed sparse matrix storage format and the algorithms for each of the solver phases are also described. The results of numerical tests of the solver on large real-world finite element problems are presented and its performance is compared to a frontal solver and the PARDISO sparse direct solver.

Finite Element Method in Density Functional Theory Electronic Structure Calculations

We summarize an ab-initio real-space approach to electronic structure calculations based on the finite-element method. This approach brings a new quality to solving Kohn Sham equations, calculating electronic states, total energy, Hellmann–Feynman forces and material properties particularly for non-crystalline, non-periodic structures. Precise, fully non-local, environment-reflecting real-space ab-initio pseudopotentials increase the efficiency by treating the core-electrons separately, without imposing any kind of frozen-core approximation. Contrary to the variety of well established k-space methods that are based on Bloch’s theorem and applicable to periodic structures, we don’t assume periodicity in any respect. The main asset of the present approach is the efficient combination of an excellent convergence control of standard, universal basis of industrially proved finite-element method and high precision of ab-initio pseudopotentials with applicability not restricted to periodic environment.

Convergence study of isogeometric analysis based on Bézier extraction in electronic structure calculations

Behavior of various, even hypothetical, materials can be predicted via ab-initio electronic structure calculations providing all the necessary information: the total energy of the system and its derivatives. In case of non-periodic structures, the existing well-established methods for electronic structure calculations either use special bases, predetermining and limiting the shapes of wave functions, or require artificial computationally expensive arrangements, like large supercells. We developed a new method for non-periodic electronic structures based on the density functional theory, environment-reflecting pseudopotentials and the isogeometric analysis with Bezier extraction, ensuring continuity for all quantities up to the second derivative. The approach is especially suitable for calculating the total energy derivatives and for molecular-dynamics simulations. Its main assets are the universal basis with the excellent convergence control and the capability to calculate precisely the non-periodic structures even lacking in charge neutrality. Within the present paper, convergence study for isogeometric analysis vs. standard finite-element approach is carried out and illustrated on sub-problems that appear in our electronic structure calculations method: the Poisson problem, the generalized eigenvalue problem and the density functional theory Kohn–Sham equations applied to a benchmark problem.

ESTIMATION OF STABILITY LIMIT BASED ON GERSHGORIN'S THEOREM FOR EXPLICIT CONTACT-IMPACT ANALYSIS SIGNORINI PROBLEM USING BIPENALTY APPROACH

Abstract. The stability properties of the bipenalty method presented in Reference [4] is studied in application to one-dimensional bipenalized Signorini problem. The attention has been paid on the critical Courant numbers estimation based on Gershgorin’s theorem. It is shown that Gershgorin’s formula overestimates maximum eigenfrequency for all penalty ratios with exception of the critical penalty ratio. Thus, smaller safer values of critical Courant numbers are obtained in comparison with exact ones calculated from the solution of eigenvalue problem.

STUDIES IN NUMERICAL STABILITY OF EXPLICIT CONTACT-IMPACT ALGORITHM TO THE FINITE ELEMENT SOLUTION OF WAVE PROPAGATION PROBLEMS

Abstract. In dynamic transient analysis, recent comprehensive studies have shown that using mass penalty together with standard stiffness penalty, the so-called bipenalty technique, preserves the critical time step in conditionally stable time integration schemes. In this paper, the bipenalty approach is applied in the explicit contact-impact algorithm based on the pre-discretization penalty formulation. The attention is focused on the stability of this algorithm. Specifically, the upper estimation of the stable Courant number on the stiffness and mass penalty is derived based on the simple dynamic system with two degrees-of-freedom. The results are verified by means of the dynamic Signorini problem, which is represented by the motion of a bar that comes into contact with a rigid obstacle.