Continuum modelling using the discrete element method. theory and implementation in an object-oriented software platform

The Discrete Element Method is a relatively new technique that has nowadays and intense research in the field of numerical methods. In its first conception, the method was designed for simulations of dynamic system of particles where each element is considered to be an independent and non-deformable entity that interacts with other particles by the laws of the contact mechanics and moves following the second Newton’s law. This first approach for the DEM has obtained excellent results for granular media simulations or another discontinuouslike case. The existing challenge nowadays for the DEM is to be able to simulate the behaviour on a continuous media discretized by a mesh of particles ruled by the equations of the DEM. Although there exist more adequate methods to solve the continuous problem as they are the different variants of the Finite Element Method, the DEM is expected to have a better behaviour when the failure of the media occurs; in terms of tracking the evolution of the fracture locally between the elements of the discretization and also the post-fractural behaviour of the material. Nowadays, there are several DEM codes that try to solve this problem although there is no one which can assure an accurate solution applicable universally to any case. The objective of the present work is to develop calculation software for the Discrete Element Method included in the platform for numerical methods KRATOS, which is developed in CIMNE. One of the goals of the so-called DEM-Application is to be able to reproduce a wide set of engineering problems; not only the discrete ones such as the excavation or agroalimentary applications but also to reproduce the continuous media, simulating compression test for concrete or asphalt samples for instance. In addition it is desired that the application permits the coupling with another methods, particularly the Finite Element Method. In order to do this, the present work includes the study of all the advances and ideas that, globally in the numerical method field and particularly in CIMNE, have been discussed to give other approaches and to keep improving and developing the to the Discrete Element Method

Formation of structural matrices for finite elements of piezoceramic structures

This paper deals with the description of a theoretical background of systematic computer algebra methods for the formation of structural matrices of piezoceramic finite elements. Piezoceramic actuators are widely used for high-precision mechanical systems such as positioning devices, manipulating systems, control equipment, etc. In this paper, the efficiency of computer algebra application was compared with the numerical integration methods of formation of the structural matrices of the finite elements. Two popular finite elements are discussed for modeling piezoceramic actuators: sector type and the triangular one. All structural matrices of the elements were derived using the computer algebra technique with the following automatic program code generation. Due to smaller floating point operations, the computer time economy is followed by an increased accuracy of computations, which is the most important gain achieved in many cases

A high-performance boundary element method and its applications in engineering

As a semi-numerical and semi-analytical method, owing to the inherent advantage, of boundary-only discretisation, the boundary element method (BEM) has been widely applied to problems with complicated geometries, stress concentration problems, infinite domain problems, and many others. However, domain integrals and non-symmetrical and dense matrix systems are two obstacles for BEM which have hindered the its further development and application. This thesis is aimed at proposing a high-performance BEM to tackle the above two drawbacks and broaden the application scope of BEM. In this thesis, a detailed introduction to the traditional BEM is given and several popular algorithms are introduced or proposed to enhance the performance of BEM. Numerical examples in heat conduction analysis, thermoelastic analysis and thermoelastic fracture problems are performed to assess the efficiency and correction of the algorithms. In addition, necessary theoretical derivations are embraced for establishing novel boundary integral equations (BIEs) for specific engineering problems. The following three parts are the main content of this thesis. (1) The first part (Part II consisting of two chapters) is aimed at heat conduction analysis by BEM. The coefficient matrix of equations formed by BEM in solving problems is fully-populated which occupy large computer memory. To deal with that, the fast multipole method (FMM) is introduced to energize the line integration boundary element method (LIBEM) to performs better in efficiency. In addition, to compute domain integrals with known or unknown integrand functions which are caused by heat sources or heterogeneity, a novel BEM, the adaptive orthogonal interpolation moving least squares (AOIMLS) method enhanced LIBEM, which also inherits the advantage of boundary-only discretisation, is proposed. Unlike LIBEM, which is an accurate and stable method for computing domain integrals, but only works when the mathematical expression of integral function in domain integrals is known, the AOIMLS enhanced LIBEM can compute domain integrals with known or unknown integral functions, which ensures all the nonlinear and nonhomogeneous problems can be solved without domain discretisation. In addition, the AOIMLS can adaptively avoid singular or ill-conditioned moment matrices, thus ensuring the stability of the calculation results. (2) In the second part (Part III consisting of four chapters), the thermoelastic problems and fracture problems are the main objectives. Due to considering thermal loads, domain integrals appear in the BIEs of the thermoelastic problems, and the expression of integrand functions is known or not depending on the temperature distribution given or not, the AOIMLS enhanced LIBEM is introduced to conduct thermoelasticity analysis thereby. Besides, a series of novel unified boundary integral equations based on BEM and DDM are derived for solving fracture problems and thermoelastic fracture problems in finite and infinite domains. Two sets of unified BIEs are derived for fracture problems in finite and infinite domains based on the direct BEM and DDM respectively, which can provide accurate and stable results. Another two sets of BIEs are addressed by employing indirect BEM and DDM, which cannot ensure a stable result, thereby a modified indirect BEM is proposed which performs much more stable. Moreover, a set of novel BIEs based on the direct BEM and DDM for cracked domains under thermal stress is proposed. (3) In the third part (Part IV consisting of one chapter), a high-efficiency combined BEM and discrete element method (DEM) is proposed to compute the inner stress distribution and particle breakage of particle assemblies based on the solution mapping scheme. For the stress field computation of particles with similar geometry, a template particle is used as the representative particle, so that only the related coefficient matrices of one template particle in the local coordinate system are needed to be calculated, while the coefficient matrices of the other particles, can be obtained by mapping between the local and global coordinate systems. Thus, the combined BEM and DEM is much more effective when modelling a large-scale particle system with a small number of distinct possible particle shapes. Furthermore, with the help of the Hoek-Brown criterion, the possible cracks or breakage paths of a particle can be obtained

Analysis of crack growth problems using the object-oriented program bemcracker2D

This paper presents an application of the boundary element method to the analysis of crack growth problems in linear elastic fracture mechanics and the correlation of results with experimental data. The methodology consists of computing stress intensity factors (SIFs), the crack growth path and the estimation of fatigue life, via an incremental analysis of the crack extension, considering two independent boundary integral equations, the displacement and traction integral equations. Moreover, a special purpose educational program for simulating two-dimensional crack growth based on the dual boundary element method (DBEM), named BemCracker2D, written in C++ with a MATLAB graphic user interface, has been developed and used to verify the adopted methodology. The numerical results are compared with those of the finite element method (FEM) and correlated with experimental data of fatigue crack-growth tests for two-dimensional structural components under simple loading, aiming to demonstrate the accuracy and efficiency of the methodology adopted, as well as to evaluate the robustness of the BemCracker2D code

Analysis of crack growth problems using the object-oriented program bemcracker2D

This paper presents an application of the boundary element method to the analysis of crack growth problems in linear elastic fracture mechanics and the correlation of results with experimental data. The methodology consists of computing stress intensity factors (SIFs), the crack growth path and the estimation of fatigue life, via an incremental analysis of the crack extension, considering two independent boundary integral equations, the displacement and traction integral equations. Moreover, a special purpose educational program for simulating two-dimensional crack growth based on the dual boundary element method (DBEM), named BemCracker2D, written in C++ with a MATLAB graphic user interface, has been developed and used to verify the adopted methodology. The numerical results are compared with those of the finite element method (FEM) and correlated with experimental data of fatigue crack-growth tests for two-dimensional structural components under simple loading, aiming to demonstrate the accuracy and efficiency of the methodology adopted, as well as to evaluate the robustness of the BemCracker2D code

Efficient domain decomposition algorithms for the solution of the helmholtz equation

The purpose of this thesis is to formulate and investigate new iterative methods for the solution of scattering problems based on the domain decomposition approach. This work is divided into three parts. In the first part, a new domain decomposition method for the perfectly matched layer system of equations is presented. Analysis of a simple model problem shows that the convergence of the new algorithm is guaranteed provided that a non-local, square-root transmission operator is used. For efficiency, in practical simulations such operators need to be localized. Current, state of the art domain decomposition algorithms use the localization technique based on rational approximation of the symbol of the transmission operator. However, the original formulation of the procedure assumed decompositions that contain no cross-points and consequently could not be used in the cross-point algorithm. In the context of the perfectly matched layer problem, we adapt the cross-point technique and combined with the rational approximation of the square root transmission operator to yield an effective algorithm. Furthermore, to reduce Krylov subspace iterations, we present a new, adequate and efficient preconditioner for the perfectly matched layer problem. The new, zero frequency limit preconditioner shows great reduction in the required number of iterations while being extremely easy to construct. In the second part of the thesis, a new domain decomposition algorithm is considered. From theoretical point of view, its formulation guarantees well-posedness of local problems. Its practicality on the other hand is evident from its efficiency and ease of implementations as compared with other, state of the art domain decomposition approaches. Moreover, the method exhibits robustness with respect to the problem frequency and is suitable for large scale simulations on a parallel computer. Finally, the third part of the thesis presents an extensible, object oriented architecture that supports development of parallel domain decomposition algorithms where local problems are solved by the finite element method. The design hides mesh implementation details and is capable of supporting various families of finite elements together with quadrature formulas of suitable degree of precision

Modified mass-spring system for physically based deformation modeling

Mass-spring systems are considered the simplest and most intuitive of all deformable models. They are computationally efficient, and can handle large deformations with ease. But they suffer several intrinsic limitations. In this book a modified mass-spring system for physically based deformation modeling that addresses the limitations and solves them elegantly is presented. Several implementations in modeling breast mechanics, heart mechanics and for elastic images registration are presented