Anton Tkachuk

Abstract

Large statically indeterminate truss and frame structures exhibit complex load-bearing behavior, and redundancy matrices are helpful for their analysis and design. Depending on the task, the full redundancy matrix or only its diagonal entries are required. The standard computation procedure has a high computational effort. Many structures fall in the category of moderately redundant, i.e., the ratio of the statical indeterminacy to the number of all load-carrying modes of all elements is less one half. This paper proposes a closed-form expression for redundancy contributions that is computationally efficient for moderately redundant systems. The expression is derived via a factorization of the redundancy matrix that is based on singular value decomposition. Several examples illustrate the behavior of the method for increasing size of systems and, where applicable, for increasing degree of statical indeterminacy.

Abstract

In the present paper, theoretical foundations of redundancy in spatially continuous, elastostatic, and linear representations of structures are derived. Adopting an operator-theoretical perspective, the redundancy operator is introduced, inspired by the concept of redundancy matrices, previously described for spatially discrete representations of structures. Studying symmetry, trace, rank, and spectral properties of this operator as well as revealing the relation to the concept of statical indeterminacy, a continuum-mechanical theory of redundancy is proposed. Here, the notion “continuum-mechanical” refers to the representation being spatially continuous. Apart from the theory itself, the novel outcome is a clear concept providing information on the distribution of statical indeterminacy in space and with respect to different load carrying mechanisms. The theoretical findings are confirmed and illustrated within exemplary rod, plane beam, and plane frame structures. The additional insight into the load carrying behavior may be valuable in numerous applications, including robust design of structures, quantification of imperfection sensitivity, assessment of adaptability, as well as actuator placement and optimized control in adaptive structures.

Abstract

In this paper, we study the accuracy and robustness of the computational time reversal approach based on the explicit finite element method for application in nondestructive testing in solids. The main goal of this paper is to find a methodology for correct and accurate reconstruction of the original source time history. For numerical modeling of frontal (forward) and reverse (backward) problems of elastic wave propagation, we use the finite element method and explicit time integration with the lumped mass matrix. The suggested methodology is applicable in each finite element open source or commercial software. A special attention is paid to prescription of boundary conditions/loading for the reverse problem for accurate reconstruction of time history of the original source. For evaluation of the reconstruction quality, we suggest certain cost functions. Based on several numerical tests, we show effects of prescription of boundary conditions/loading in time reversal, effect of mesh size and time step size, an unknown obstacle, a number of sources, and environmental disturbance (noise) on the correctness of reconstruction of the original source.

Abstract

Customization of finite elements for low dispersion error through grid dispersion analysis requires a symbolic expansion of a determinant of a representative dynamic stiffness matrix. Such an expansion turns out to be a bottleneck for many practical cases with the size of the representative matrix greater than eight or ten even if the modern computer algebra systems are applied. In this contribution, we propose an alternative approach for low-dispersion customization that avoids explicit determinant expansion. This approach reduces the customization problem to a series of quadratic programming problems and consist of two main steps. Firstly, the customization problem is reformulated as a rank minimization problem for the representative dynamic stiffness matrix evaluated at several discrete pairs of wavenumbers and frequencies. Secondly, the rank minimization problem is solved approximately via log-det heuristic. Examples for customization of reciprocal mass matrices illustrate capabilities of the proposed approach.

Abstract

Finite elements with Allman’s rotations provide good computational efficiency for explicit codes exhibiting less locking than linear elements and lower computational cost than quadratic finite elements. One way to further raise their efficiency is to increase the feasible time step or increase the accuracy of the lowest eigenfrequencies via reciprocal mass matrices. This paper presents a formulation for variationally scaled reciprocal mass matrices and an efficient estimator for the feasible time step for finite elements with Allman’s rotations. These developments take special care of two core features of such elements: existence of spurious zero‐energy rotation modes implying the incompleteness of the ansatz spaces, and the presence of mixeddimensional degrees of freedom. The former feature excludes construction of dual bases used in the standard variational derivation of reciprocal mass matrices. The latter feature destroys the efficiency of the existing nodal‐based time step estimators stemming from the Gershgorin’s eigenvalue bound. Finally, the developments are tested for standard benchmarks and triangular, quadrilateral and tetrahedral finite elements with Allman’s rotations.

Abstract

In this contribution, a novel local, node-based time step estimate for reciprocal mass matrices is proposed. Element-based estimates turn out to be not generally conservative and are consequently inadequate. Therefore, the nodal time step estimate for diagonally lumped mass matrices based on Gershgorin’s theorem is further developed for application to reciprocal mass matrices. Additionally, simplifications of the proposed time step estimate that improve computational efficiency, especially for contact problems with the penalty method, are discussed and evaluated by numerical examples.

Abstract

The stability and reflection-transmission properties of the bipenalty method are studied in application to explicit finite element analysis of one-dimensional contact-impact problems. It is known that the standard penalty method, where an additional stiffness term corresponding to contact boundary conditions is applied, attacks the stability limit of finite element model. Generally, the critical time step size rapidly decreases with increasing penalty stiffness. Recent comprehensive studies have shown that the so-called bipenalty technique, using mass penalty together with standard stiffness penalty, preserves the critical time step size associated to contact-free bodies. In this paper, the influence of the penalty ratio (ratio of stiffness and mass penalty parameters) on stability and reflection-transmission properties in one-dimensional contact-impact problems using the same material and mesh size for both domains is studied. The paper closes with numerical examples, which demonstrate the stability and reflection-transmission behavior of the bipenalty method in one-dimensional contact-impact and wave propagation problems of homogeneous materials.

Abstract

In this contribution, variationally consistent inertia templates for B-spline and NURBS-based finite elements are proposed as a unified concept for two different purposes: Customization of the template allows construction of masses and reciprocal masses with desired properties like higher-order accuracy or improved dispersion behavior; Inertia scaling allows substantial speed-up for explicit dynamics by increased critical time steps. The derivation of the template is based on a three-field parametrized functional as in previous works of the authors’ group, but with modified primary variables, namely displacement, velocity and mass-specific linear momentum. The latter allows for mass-preservation for non-constant density throughout the domain and is therefore an enhancement to the formulation proposed in Tkachuk and Bischoff (2015). With the focus on B-spline and NURBS-based finite elements, the proposed template provides alternatives to the row-sum- lumped mass matrix, which is only 2nd order accurate independent of the polynomial order. Earlier proposed algebraically constructed higher order masses from the literature can be reconstructed in the variational setting described here as special instances. Furthermore, higher-order reciprocal masses can be constructed from the template. They are especially attractive for explicit dynamics as there is no extra expense per time step compared with lumped mass for linear problems. For non-linear problems only small overhead is expected, but this paper focuses on linear problems only and mainly undistorted meshes. Tuning the method towards inertia scaling, a reduction of the maximum eigenfrequency by 25%–40% is obtained in the examples herein, whereas the accuracy is higher than for lumped or consistent mass.

Abstract

Purpose Current procedures of buckling load estimation for thin-walled structures may provide very conservative estimates. Their refinement offers the potential to use structure and material properties more efficiently. Due to the large variety of design variables, for example laminate layup in composite structures, a prohibitively large number of tests would be required for experimental assessment, and thus reliable numerical techniques are of particular interest. The purpose of this paper is to analyze different methods of numerical buckling load estimation, formulate simulation procedures suitable for commercial software and give recommendations regarding their application. All investigations have been carried out for cylindrical composite shells; however similar approaches are feasible for other structures as well. Design/methodology/approach We develop a concept to apply artificial load imperfections with the aim to estimate as good as possible lower bounds for the buckling loads of shells for which the actual physical imperfections are not known. Single and triple perturbation load approach, global and local dynamic perturbation approach and path following techniques are applied to the analysis of a cylindrical composite shell with known buckling characteristics. Results of simulations are compared with published experimental data. Findings A single perturbation load approach is reproduced and modified. Buckling behavior for negative values of the perturbation load is examined and a pattern similar to a positive perturbation load is observed. Simulations with three perturbation forces show a decreased (i. e. more critical) value of the buckling load compared to the single perturbation load approach. Global and local dynamic perturbation approaches exhibit a behavior suitable for lower bound estimation for structures with arbitrary geometries. Originality/value Various load imperfection approaches to buckling load estimation are validated and compared. All investigated methods do not require knowledge of the real geometrical imperfections of the structure. Simulations were performed using a commercial finite element code. Investigations of sensitivity with respect to a single perturbation load are extended to the negative range of the perturbation load amplitude. A specific pattern for a global perturbation approach was developed, and based on it a novel simulation procedure is proposed.

Abstract

Multimaterial polymer printers allow the placement of different material phases within a composite, where some or all of the materials may exhibit an active response. Utilizing the shape memory (SM) behavior of at least one of the material phases, active composites can be three-dimensional (3D) printed such that they deform from an initially flat plate into a curved structure. This paper introduces a topology optimization approach for finding the spatial arrangement of shape memory polymers (SMPs) within a passive matrix such that the composite assumes a target shape. The optimization approach combines a level set method (LSM) for describing the material layout and a generalized formulation of the extended finite-element method (XFEM) for predicting the response of the printed active composite (PAC). This combination of methods yields optimization results that can be directly printed without the need for additional postprocessing steps. Two multiphysics PAC models are introduced to describe the response of the composite. The models differ in the level of accuracy in approximating the residual strains generated by a thermomechanical programing process. Comparing XFEM predictions of the two PAC models against experimental results suggests that the models are sufficiently accurate for design purposes. The proposed optimization method is studied with examples where the target shapes correspond to a plate-bending type deformation and to a localized deformation. The optimized designs are 3D printed and the XFEM predictions are compared against experimental measurements. The design studies demonstrate the ability of the proposed optimization method to yield a crisp and highly resolved description of the optimized material layout that can be realized by 3D printing. As the complexity of the target shape increases, the optimal spatial arrangement of the material phases becomes less intuitive, highlighting the advantages of the proposed optimization method.

Abstract

Classical explicit finite element formulations rely on lumped mass matrices. A diagonalized mass matrix enables a trivial computation of the acceleration vector from the force vector. Recently, non-diagonal mass matrices for explicit finite element analysis (FEA) have received attention due to the selective mass scaling (SMS) technique. SMS allows larger time step sizes without substantial loss of accuracy. However, an expensive solution for accelerations is required at each time step. In the present study, this problem is solved by directly constructing the inverse mass matrix. First, a consistent and sparse inverse mass matrix is built from the modified Hamiltons principle with independent displacement and momentum variables. Usage of biorthogonal bases for momentum allows elimination of momentum unknowns without matrix inversions and directly yields the inverse mass matrix denoted here as reciprocal mass matrix (RMM). Secondly, a variational mass scaling technique is applied to the RMM. It is based on the penalized Hamiltons principle with an additional velocity variable and a free parameter. Using element-wise bases for velocity and a local elimination yields variationally scaled RMM. Thirdly, examples illustrating the efficiency of the proposed method for simplex elements are presented and discussed.

Abstract

The problem of optimal selective mass scaling for linearized elasto-dynamics is discussed. Optimal selective mass scaling should provide solutions for dynamical problems that are close to the ones obtained with a lumped mass matrix, but at much smaller computational costs. It should be equally applicable to all structurally relevant load cases. The three main optimality criteria, namely eigenmode preservation, small number of non-zero entries and good conditioning of the mass matrix are explicitly formulated in the article. An example of optimal mass scaling which relies on redistribution of mass on a global system level is constructed. Alternative local mass scaling strategies are proposed and compared with existing methods using one modal and two transient numerical examples.

Abstract

A new variational method for selective mass scaling is proposed. It is based on a new penalized Hamilton’s principle where relations between variables for displacement, velocity and momentum are imposed via a penalty method. Independent spatial discretization of the variables along with a local static condensation for velocity and momentum yields a parametric family of consistent mass matrices. In this framework new mass matrices with desired properties can be constructed. It is demonstrated how usage of these non-diagonal mass matrices decreases the maximum frequency of the discretized system and allows for larger steps in explicit time integration. At the same time the lowest eigenfrequencies in the range of interest and global structural response are not significantly changed. Results of numerical experiments for two-dimensional and three-dimensional problems are discussed.

Abstract

An alternative spatial semi-discretization of dynamic contact based on a modified Hamilton’s principle is proposed. The modified Hamilton’s principle uses displacement, velocity and momentum as variables, which allows their independent spatial discretization. Along with a local static condensation for velocity and momentum, it leads to an approach with a hybrid-mixed consistent mass matrix. An attractive feature of such a formulation is the possibility to construct hybrid singular mass matrices with zero components at those nodes where contact is collocated. This improves numerical stability of the semi-discrete problem: the differential index of the underlying differential-algebraic system is reduced from 3 to 1, and spurious oscillations in the contact pressure, which are commonly reported for formulations with Lagrange multipliers, are significantly reduced. Results of numerical experiments for truss and Timoshenko beam elements are discussed. In addition, the properties of the novel discretization scheme for an unconstrained dynamic problem are assessed by a dispersion analysis.

Abstract

In this contribution, a novel local, node-based time step estimate for reciprocal mass matrices with improved sharpness is proposed. Reciprocal mass matrices are sparse matrices used in explicit dynamics that allow computation of the nodal acceleration from the total force vector. They aim for higher critical time steps or/and accuracy than the lumped mass matrix approximation. Since the eigenvalue inequality by Fried does not hold for reciprocal mass matrices, element-based estimates may be not conservative and are consequently inadequate. Therefore, nodal time step estimates are preferred for reciprocal mass matrices. A nodal time step estimate requires an eigenvalue bound using row or/and column information associated with the nodal degrees of freedom. Recently, an efficient estimate was proposed that uses Gershgorin’s circle theorem and a corresponding bound. This estimate is always conservative but it usually leaves a 10-30% gap to the true eigenvalue. This raises the question whether a sharper estimate may be constructed by using a different eigenvalue bound. A recent paper by Cottereau and Sevilla compares five eigenvalue bounds for a diagonal mass matrix with spectral finite elements and indicates that the Ostrowski’s bound is the sharpest bound for most of the considered cases. In this contribution, the latter estimate is further developed for the case of reciprocal mass matrices and an extra improvement via similarity transformation for elements with rotary degrees of freedom is discussed. Finally, the estimate is tested for a set of common finite elements in explicit dynamics.

Abstract

Um dynamische Finite-Elemente-Berechnungen mit expliziter Zeitintegration zu beschleunigen, ist es in der Praxis üblich, Massenskalierung anzuwenden. Damit soll die kritische Zeitschrittweite erhöht werden ohne maßgeblich an Genauigkeit in den entscheidenden niederfrequenten Moden zu verlieren. Bei der konventionellen Massenskalierung (CMS) wird künstliche Masse zu den Diagonaltermen der diagonalisierten Massenmatrix addiert und die Diagonalform der Massenmatrix bleibt erhalten. CMS wird gewöhnlich auf eine geringe Anzahl kleiner oder steifer Elemente angewandt, deren hohe Eigenfrequenzen die kritische Zeitschrittweite begrenzen. Dabei wird allerdings die Trägheit der Struktur erhöht, was zu unphysikalischen Effekten führen kann. Mit einer sogenannten selektiven Massenskalierung kann wenigstens die translatorische Trägheit (d.h. der Impuls bei gleichmäßiger Anfangsgeschwindigkeit) erhalten werden, allerdings auf Kosten von Nebendiagonaltermen in der Massenmatrix. Diese Methode lässt sich gleichmäßig auf die gesamte Struktur anwenden und hat geringere unphysikalische Nebeneffekte. Bislang werden skalierte Massenmatrizen rein algebraisch, z.B. steifigkeitsproportional konstruiert. Die Auswahl an Skalierungs-Templates ist beschränkt und ihnen liegt keine konsistente Formulierung zugrunde. In diesem Artikel werden variationelle Methoden zur selektiven Massenskalierung vorgestellt, die nicht nur mathematisch konsistent sind, sondern es auch erlauben, gezielt bestimmte Eigenschaften wie die Erhaltung der Trägheit, Reduktion der höchsten Frequenzen und Genauigkeit bei den niedrigen Frequenzen einzustellen. Die Leistungsfähigkeit der Methode wird an numerischen Beispielen mit Tetraeder- und Hexaeder-Elementen demonstriert.

Abstract

For practical application in engineering numerical simulations are required to be reliable and reproducible. Unfortunately crash simulations are highly complex and nonlinear and small changes in the initial state can produce big changes in the results. This is caused partially by physical instabilities and partially by numerical instabilities. Aim of the project is to identify the numerical sensitivities in crash simulations and suggest methods to reduce the scatter of the results.

Abstract

A Lagrange multipliers formulation for dynamical frictionless thermo-elastic contact problem is considered. Thermal deformations and dependency of contact thermal resistance on contact pressure are assumed to be the only two coupling effects. Application of standard Newmark method to the problem may lead to spurious numerical oscillations of contact pressures and heat fluxes, inaccurate or divergent solutions. A modification of the Newmark method is proposed where contact contributions are integrated non-monolithically with backward Euler. Elimination of spurious numerical oscillations is shown in a numerical example.

Abstract

For practical application in engineering numerical simulations are required to be reliable and reproducible. Unfortunately crash simulations are highly complex and nonlinear and small changes in the initial state can produce big changes in the results. This is caused partially by physical instabilities and partially by numerical instabilities. Aim of the project is to identify the numerical sensitivities in crash simulations and suggest methods to reduce the scatter of the results.

Abstract

For practical application in engineering numerical simulations are required to be reliable and reproducible. Unfortunately crash simulations are highly complex and nonlinear and small changes in the initial state can produce big changes in the results. This is caused partially by physical instabilities and partially by numerical instabilities. Aim of the project is to identify the numerical sensitivities in crash simulations and suggest methods to reduce the scatter of the results. Work has been undertaken to identify sources of sensitivities through parameter studies, to improve existing mathematical formulations, e.g. of contact-impact and material models. Furthermore a tool is developed for generation of code from mathematical descriptions of finite elements with the aim to reduce effort required to make modifications of implementations of FEM models.

Abstract

Singular and selectively-scaled mass matricesare useful for finite element modeling of numerous problems of structural dynamics, for example for low velocity impact, deep drawing and drop test simulations. Singular mass matrices allow significant reduction of spurious temporal oscillations of contact pressure. The application of selective mass scaling in the context of explicit dynamics reduces the computational costs without substantial loss in accuracy. Known methods for singular and selectively-scaled mass matrices rely on special quadrature rules or algebraic manipulations applied on the standard mass matrices. This thesis is dedicated to variationally rigorous derivation and analysis of these alternative matrices. The theoretical basis of this thesis is a novel parametric HAMILTON’s principle with independent variables for displacement, velocity and momentum. The numerical basis is hybrid-mixed discretization of the novel mixed principle and skillful tuning of ansatz spaces and free parameters. The qualities of novel mass matrices are thoroughly analyzed by various tests and benchmarks. The thesis has three main parts. In the first part of the thesis, the essential fundamentals and notations are introduced. This includes the basic continuum mechanics, the local form of an initial boundary value problem for an elasto-dynamic contact problem and its treatment with finite elements. In addition, an extension of the central difference method to non-diagonal mass matrices and a theoretical estimate of speed-up with selective mass scaling is given. Besides, a motivation for implementation of alternative mass matrices is given. In the second part of the thesis, the novel variational approach for elasto-dynamic problems is presented. The corner stone of the thesis is the derivation of the novel penalized HAMILTON’s principle and an extension of the modified HAMILTON’s principle for small sliding unilateral contact. These formulations are discretized in space with the BUBNOV-GALERKIN approach. As a result, families of singular and selectively-scaled mass matrices are obtained. The correspondingshape functions are builtfor several families of finite elements. These families include truss and TIMOSHENKO beam elements for one-dimensional problems, as well as solid elements for two and three dimensions. Shape functions for singular mass matrices are derived for quadratic and cubic elements. Selectively-scaled mass matrices are given for elements up to the order three. In the third part of the thesis, the novel mass matrices are analyzed and an outlook for future work is given. Propagation of harmonic waves, free and forced vibrations and impact problems are used for evaluation of the new mass matrices. First, the propagation of harmonic waves is studied with the help of a FOURIER analysis applied to the semi-discretized equation of motion. This analysis results in a set of dispersion relations. Comparison of the analytical expressions for discrete dispersion relations with the corresponding continuous ones allows efficient error estimation. In this way, the proposed truss and beam elements are analyzed. Secondly, eigenvalue problems are solved for two-and three-dimensional problems. The error in the lowest frequencies (modes) and in the whole spectrum is computed. Thirdly, spectral response curves for forced vibrations are obtained for the new mass matrices in ranges of interest. These curves are compared with the ones obtained with consistent mass matrices via the frequency response assurance criterion. The values of the frequency response assurance criterion indicate the error for linear problems. Finally, several transient examples are solved with singular and scaled mass matrices. These examples confirm expected superiority of singular mass matrices for impact problems, i.e. spurious temporal oscillations of contact pressures are significantly reduced. Variational selective mass scaling reduces computational cost of explicit dynamic simulations. In the outlook, possible developments regarding new element types, alternative weak forms and several multi-physic applications are proposed. As by-product of this thesis, patch tests for inertia terms, an overview of parametric and non-parametric variational principles of elasto-dynamics and a derivation of the penalized HAMILTON’s principle with a semi-inverse method can be noted. Besides, the topic of finite element technology for mass matrices is posed. This can open new horizons for evolving branches of computational dynamics such as drop test and car crash simulations, phononic crystals and devices.