Radek Kolman

Using finite element method for the determination of elastic moduli by resonant ultrasound spectroscopy

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. Presently, Levenberg–Marquardt’s (LM) algorithm with the Ritz method for the solution of eigenfrequencies is usually applied. The LM method is based on the modified Newton–Raphson procedure, where all the eigenfrequencies and eigenvectors must be known for the computation of relevant gradients. Finite element method offers an analogous but more general optimization. The tested specimen, for instance, can be made of a composite material consisting of several layers with different material properties; the form of the specimen can be of a more complex shape, etc. In the present work, the elastic moduli are optimized by the fixed point iteration method, which r...

Dispersion of elastic waves in the contact-impact problem of a long cylinder

Numerical dispersion of two-dimensional finite elements was studied. The outcome of the dispersion study was verified by the numerical and analytical solutions to the longitudinal impact of two long cylindrical bars. In accordance with the results of the dispersion analysis it was demonstrated that the quadratic elements showed better accuracy than the linear ones.

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.

Graphene under direct compression: Stress effects and interlayer coupling

In this work we explore mechanical properties of graphene samples of variable thickness. For this purpose, we coupled a high pressure sapphire anvil cell to a micro-Raman spectrometer. From the evolution of the G band frequency with stress we document the importance the substrate has on the mechanical response of graphene. On the other hand, the appearance of disorder as a conse-quence of the stress treatment has a negligible effect on the high stress behaviour of graphene.

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.

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.

INVERSE MASS MATRIX VIA THE METHOD OF LOCALIZED LAGRANGE MULTIPLIERS

INVERSE MASS MATRIX VIA THE METHOD OF LOCALIZED LAGRANGE MULTIPLIERS José A. González, R. Kolman, S.S. Cho, C. A. Felippa, K.C. Park 1Escuela Técnica Superior de Ingenierı́a, Universidad de Sevilla Camino de los Descubrimientos s/n, Seville E-41092, Spain e-mail: japerez@us.es 2Institute of Thermomechanics, The Czech Academy of Sciences Dolejškova 5, 182 00 Prague, Czech Republic e-mail: kolman@it.cas.cz 3Reactor Mechanical Engineering Division, Korea Atomic Energy Research Institute 999-111 Daedeok-Daero, Yuseong-gu, Daejeon 305-353, Republic of Korea e-mail: sscho96@kaeri.re.kr 4Department of Aerospace Engineering Sciences University of Colorado at Boulder, CO 80309-429, USA e-mail: {carlos.felippa,kcpark}@colorado.edu

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.

Sample Geometry and the Brittle-Ductile Behavior of Edge Cracks in 3D Atomistic Simulations by Molecular Dynamics

We present new results of molecular dynamic (MD) simulations in 3D bcc iron crystals with edge cracks (001)[010] and (-110)[110] loaded in mode I. Different sample geometries of SEN type were tested with negative and positive values of T-stress according to continuum prediction by Fett.

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.