267 research outputs found
A Study of Speed of the Boundary Element Method as applied to the Realtime Computational Simulation of Biological Organs
In this work, possibility of simulating biological organs in realtime using
the Boundary Element Method (BEM) is investigated. Biological organs are
assumed to follow linear elastostatic material behavior, and constant boundary
element is the element type used. First, a Graphics Processing Unit (GPU) is
used to speed up the BEM computations to achieve the realtime performance.
Next, instead of the GPU, a computer cluster is used. Results indicate that BEM
is fast enough to provide for realtime graphics if biological organs are
assumed to follow linear elastostatic material behavior. Although the present
work does not conduct any simulation using nonlinear material models, results
from using the linear elastostatic material model imply that it would be
difficult to obtain realtime performance if highly nonlinear material models
that properly characterize biological organs are used. Although the use of BEM
for the simulation of biological organs is not new, the results presented in
the present study are not found elsewhere in the literature.Comment: preprint, draft, 2 tables, 47 references, 7 files, Codes that can
solve three dimensional linear elastostatic problems using constant boundary
elements (of triangular shape) while ignoring body forces are provided as
supplementary files; codes are distributed under the MIT License in three
versions: i) MATLAB version ii) Fortran 90 version (sequential code) iii)
Fortran 90 version (parallel code
Veamy: an extensible object-oriented C++ library for the virtual element method
This paper summarizes the development of Veamy, an object-oriented C++
library for the virtual element method (VEM) on general polygonal meshes, whose
modular design is focused on its extensibility. The linear elastostatic and
Poisson problems in two dimensions have been chosen as the starting stage for
the development of this library. The theory of the VEM, upon which Veamy is
built, is presented using a notation and a terminology that resemble the
language of the finite element method (FEM) in engineering analysis. Several
examples are provided to demonstrate the usage of Veamy, and in particular, one
of them features the interaction between Veamy and the polygonal mesh generator
PolyMesher. A computational performance comparison between VEM and FEM is also
conducted. Veamy is free and open source software
Accurate computation of Galerkin double surface integrals in the 3-D boundary element method
Many boundary element integral equation kernels are based on the Green's
functions of the Laplace and Helmholtz equations in three dimensions. These
include, for example, the Laplace, Helmholtz, elasticity, Stokes, and Maxwell's
equations. Integral equation formulations lead to more compact, but dense
linear systems. These dense systems are often solved iteratively via Krylov
subspace methods, which may be accelerated via the fast multipole method. There
are advantages to Galerkin formulations for such integral equations, as they
treat problems associated with kernel singularity, and lead to symmetric and
better conditioned matrices. However, the Galerkin method requires each entry
in the system matrix to be created via the computation of a double surface
integral over one or more pairs of triangles. There are a number of
semi-analytical methods to treat these integrals, which all have some issues,
and are discussed in this paper. We present novel methods to compute all the
integrals that arise in Galerkin formulations involving kernels based on the
Laplace and Helmholtz Green's functions to any specified accuracy. Integrals
involving completely geometrically separated triangles are non-singular and are
computed using a technique based on spherical harmonics and multipole
expansions and translations, which results in the integration of polynomial
functions over the triangles. Integrals involving cases where the triangles
have common vertices, edges, or are coincident are treated via scaling and
symmetry arguments, combined with automatic recursive geometric decomposition
of the integrals. Example results are presented, and the developed software is
available as open source
A collocated C0 finite element method: Reduced quadrature perspective, cost comparison with standard finite elements, and explicit structural dynamics
We demonstrate the potential of collocation methods for efficient higher-order analysis on standard nodal finite element meshes. We focus on a collocation method that is variationally consistent and geometrically flexible, converges optimally, embraces concepts of reduced quadrature, and leads to symmetric stiffness and diagonal consistent mass matrices. At the same time, it minimizes the evaluation cost per quadrature point, thus reducing formation and assembly effort significantly with respect to standard Galerkin finite element methods. We provide a detailed review of all components of the technology in the context of elastodynamics, that is, weighted residual formulation, nodal basis functions on Gauss–Lobatto quadrature points, and symmetrization by averaging with the ultra-weak formulation. We quantify potential gains by comparing the computational efficiency of collocated and standard finite elements in terms of basic operation counts and timings. Our results show that collocation is significantly less expensive for problems dominated by the formation and assembly effort, such as higher-order elastostatic analysis. Furthermore, we illustrate the potential of collocation for efficient higher-order explicit dynamics. Throughout this work, we advocate a straightforward implementation based on simple modifications of standard finite element codes. We also point out the close connection to spectral element methods, where many of the key ideas are already established
A one point integration rule over star convex polytopes
In this paper, the recently proposed linearly consistent one point integration rule for the meshfree methods is extended to arbitrary polytopes. The salient feature of the proposed technique is that it requires only one integration point within each n-sided polytope as opposed to 3n in Francis et al. (2017) and 13n integration points in the conventional approach for numerically integrating the weak form in two dimensions. The essence of the proposed technique is to approximate the compatible strain by a linear smoothing function and evaluate the smoothed nodal derivatives by the discrete form of the divergence theorem at the geometric center. This is done by Taylor's expansion of the weak form which facilitates the use of the smoothed nodal derivatives acting as the stabilization term. This translates to 50% and 30% reduction in the overall computational time in the two and three dimensions, respectively, whilst preserving the accuracy and the convergence rates. Th
Numerical Computation of approximate Generalized Polarization Tensors
In this paper we describe a method to compute Generalized Polarization
Tensors. These tensors are the coefficients appearing in the multipolar
expansion of the steady state voltage perturbation caused by an inhomogeneity
of constant conductivity. As an alternative to the integral equation approach,
we propose an approximate semi-algebraic method which is easy to implement.
This method has been integrated in a Myriapole, a matlab routine with a
graphical interface which makes such computations available to non-numerical
analysts
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