Euclides Mesquita

Constant Boundary Elements on graphics hardware: a GPU-CPU complementary implementation

Numerical simulation of engineering problems has reached such a large scale that the use of a parallel computing approach is required to obtain solutions within a reasonable time. Recent efforts have been made to implement these large scale computational tasks on general-purpose programmable graphics hardware (GPGPU). The Graphics Processing Unit (GPU) is specially well-suited to address problems that can be formulated in form of data-parallel computations with high arithmetic intensity. This work addresses the implementation of the direct version of the Boundary Element Method (DBEM) on a complementary GPU-CPU system. In this article, constant elements were used for the solution of 2D potential problems. A serial implementation of the BEM was rewritten under the SIMT (Single Instruction Multiple Thread) parallel programming paradigm. The code was developed on an NVidiaTM CUDA programming environment. The efficiency of the implemented strategies is investigated by solving a representative 2D potential problem. The paper reviews in detail the classical BEM formulation in order to be able to address the possible parallelization steps in the numerical implementation. The article reports the performance of the GPU-CPU system compared to the classical CPU-based system for an increasing number of boundary elements.

Numerical methods for the dynamics of unbounded domains

The present article discusses the relation between boundary conditions and the Sommerfeld radiation condition underlying the dynamics of unbounded domains. It is shown that the classical Dirichlet, Neumann and mixed boundary conditions do not fulfill the radiation condition. In the sequence, three strategies to incorporate the radiation condition in numerical methods are outlined. The inclusion of Infinite Elements in the realm of the Finite Element Method (FEM), the Dirichlet-to-Neumann (DtN) mapping and the Boundary Element Method (BEM) are described. Examples of solved dynamic problems in unbounded domains are given for the Helmholtz and the Navier operators. The advantages and limitations of the methodologies are discussed and pertinent literature is provided.

Dynamic Stationary Response of Reinforced Plates by the Boundary Element Method

A direct version of the boundary element method (BEM) is developed to model the stationary dynamic response of reinforced plate structures, such as reinforced panels in buildings, automobiles, and airplanes. The dynamic stationary fundamental solutions of thin plates and plane stress state are used to transform the governing partial differential equations into boundary integral equations (BIEs). Two sets of uncoupled BIEs are formulated, respectively, for the in-plane state (membrane) and for the out-of-plane state (bending). These uncoupled systems are joined to form a macro-element, in which membrane and bending effects are present. The association of these macro-elements is able to simulate thin-walled structures, including reinforced plate structures. In the present formulation, the BIE is discretized by continuous and/or discontinuous linear elements. Four displacement integral equations are written for every boundary node. Modal data, that is, natural frequencies and the corresponding mode shapes of reinforced plates, are obtained from information contained in the frequency response functions (FRFs). A specific example is presented to illustrate the versatility of the proposed methodology. Different configurations of the reinforcements are used to simulate simply supported and clamped boundary conditions for the plate structures. The procedure is validated by comparison with results determined by the finite element method (FEM).

Transient wave propagation phenomena at visco-elastic half-spaces under distributed surface loadings

This article analyzes the transient wave propagation phenomena that take place at 2D viscoelastic half-spaces subjected to spatially distributed surface loadings and to distinct temporal excitations. It starts with a fairly detailed review of the existing strategies to describe transient analysis for elastic and viscoelastic continua by means of the Boundary Element Method (BEM). The review explores the possibilities and limitations of the existing transient BEM procedures to describe dynamic analysis of unbounded viscoelastic domains. It proceeds to explain the strategy used by the authors of this article to synthesize numerically fundamental solutions or auxiliary states that allow an accurate analysis of transient wave propagation phenomena at the surface of viscoelastic half-spaces. In particular, segments with spatially constant and linear stress distributions over a halfspace surface are considered. The solution for the superposition of constant and discontinuous adjacent elements as well as linear and continuous stress distributions is addressed. The in uence of the temporal excitation type and duration on the transient response is investigated. The present study is based on the numerical solution of stress boundary value problems of (visco)elastodynamics. In a first stage, the solution is obtained in the frequency domain. A numerical integration strategy allows the stationary solutions to be determined for very high frequencies. The transient solutions are obtained, in a second stage, by applying the Fast Fourier Transform (FFT) algorithm to the previously synthesized frequency domain solutions. Viscoelastic effects are taken into account by means of the elastic-viscoelastic correspondence principle. By analyzing the transient solution of the stress boundary value problems, it is possible to show that from every surface stress discontinuity three wave fronts are generated. The displacement velocity of these wave fronts can be associated to compression, shear and Rayleigh waves. It is shown that the half-space transient displacement solutions present abrupt jumps or oscillations which can be correlated to the arrival of these wave fronts at the observation point. Such a detailed analysis connecting half-space transient responses to the wave propagation fronts in viscoelastic half-spaces have not been reported in the reviewed literature.

SCR-Seafloor Interaction Modeling With Winkler, Pasternak and Kerr Beam-on-Elastic-Foundation Theories

This paper discusses the use of three analytical models for the SCR-soil interaction analysis. Analytical derivations for Winkler, Pasternak and Kerr models are presented. The effects of the adopted model on displacements and stresses near the riser touchdown point are addressed. Comparisons with the results from a continuous soil model calculated with finite elements are also presented. In the regions with large displacement gradients, the Kerr model, which considers the influence of the elasticity from neighboring area, presents the best agreement with the 3D Finite Element model.© 2009 ASME

Mechanical Behavior of Nano Structures Using Atomic-Scale Finite Element Method (AFEM)

THIS WORK PRESENTS A DETAILED DESCRIPTION OF THE FORMULATION AND IM-PLEMENTATION OF THE ATOMISTIC FINITE ELEMENT METHOD AFEM, EXEMPLI-FIED IN THE ANALYSIS OF ONE- AND TWO-DIMENSIONAL ATOMIC DOMAINS GOV-ERNED BY THE LENNARD JONES INTERATOMIC POTENTIAL. THE METHODOLOGY TO SYNTHESIZE ELEMENT STIFFNESS MATRICES AND LOAD VECTORS, THE POTENTIAL ENERGY MODIFICATION OF THE ATOMISTIC FINITE ELEMENTS (AFE) TO ACCOUNT FOR BOUNDARY EDGE EFFECTS, THE INCLUSION OF BOUNDARY CONDITIONS IS CARE-FULLY DESCRIBED. THE CONCEPTUAL RELATION BETWEEN THE CUT-OFF RADIUS OF INTERATOMIC POTENTIALS AND THE NUMBER OF NODES IN THE AFE IS ADDRESSED AND EXEMPLIFIED FOR THE 1D CASE. FOR THE 1D CASE ELEMENTS WITH 3, 5 AND 7 NODES WERE ADDRESSED. THE AFEM HAS BEEN USED TO DESCRIBE THE ME-CHANICAL BEHAVIOR OF ONE-DIMENSIONAL ATOMIC ARRAYS AS WELL AS TWO-DIMENSIONAL LATTICES OF ATOMS. THE EXAMPLES ALSO INCLUDED THE ANALYSIS OF PRISTINE DOMAINS, AS WELL AS DOMAINS WITH MISSING ATOMS, DEFECTS, OR VACANCIES. RESULTS ARE COMPARED WITH CLASSICAL MOLECULAR DYNAMIC SIMULATIONS (MD) PERFORMED USING A COMMERCIAL PACKAGE. THE RESULTS HAVE BEEN VERY ENCOURAGING IN TERMS OF ACCURACY AND IN THE COMPUTA-TIONAL EFFORT NECESSARY TO EXECUTE BOTH METHODOLOGIES, AFEM AND MD. THE METHODOLOGY CAN BE EXPANDED TO MODEL ANY DOMAIN DESCRIBED BY AN INTERATOMIC ENERGY POTENTIAL.

Stationary Dynamic Displacement Solutions for a Rectangular Load Applied within a 3D Viscoelastic Isotropic Full Space—Part II: Implementation, Validation, and Numerical Results

In part I of the present article the formulation for a dynamic stationary semianalytical solution for a spatially constant load applied over a rectangular surface within a viscoelastic isotropic full-space has been presented. The solution is obtained within the frame of a double Fourier integral transform. These inverse integral transforms must be evaluated numerically. In the present paper, the technique to evaluate numerically the inverse double Fourier integrals is described. The procedure is validated, and a number of original displacement results for the stationary loading case are reported.

Stationary Dynamic Displacement Solutions for a Rectangular Load Applied within a 3D Viscoelastic Isotropic Full Space—Part I: Formulation

A dynamic stationary semianalytical solution for a spatially constant load applied over a rectangular surface within a viscoelastic isotropic full space is presented. The solution is obtained within the frame of a double Fourier integral transform. Closed-form solutions for general loadings within the full space are furnished in the transformed wave number domain. Expressions for three boundary value problems, associated to a normal and two tangential rectangular loadings in the original physical space, are given in terms of a double inverse Fourier integral. These inverse integral transforms must be evaluated numerically. In the second part of the present paper a strategy to evaluate these integrals is described, the procedure validated and a number of original results are reported.