14,796 research outputs found
Standard finite elements for the numerical resolution of the elliptic Monge-Ampere equation: Aleksandrov solutions
We prove a convergence result for a natural discretization of the Dirichlet
problem of the elliptic Monge-Ampere equation using finite dimensional spaces
of piecewise polynomial C0 or C1 functions. Standard discretizations of the
type considered in this paper have been previous analyzed in the case the
equation has a smooth solution and numerous numerical evidence of convergence
were given in the case of non smooth solutions. Our convergence result is valid
for non smooth solutions, is given in the setting of Aleksandrov solutions, and
consists in discretizing the equation in a subdomain with the boundary data
used as an approximation of the solution in the remaining part of the domain.
Our result gives a theoretical validation for the use of a non monotone finite
element method for the Monge-Amp\`ere equation
Conforming restricted Delaunay mesh generation for piecewise smooth complexes
A Frontal-Delaunay refinement algorithm for mesh generation in piecewise
smooth domains is described. Built using a restricted Delaunay framework, this
new algorithm combines a number of novel features, including: (i) an
unweighted, conforming restricted Delaunay representation for domains specified
as a (non-manifold) collection of piecewise smooth surface patches and curve
segments, (ii) a protection strategy for domains containing curve segments that
subtend sharply acute angles, and (iii) a new class of off-centre refinement
rules designed to achieve high-quality point-placement along embedded curve
features. Experimental comparisons show that the new Frontal-Delaunay algorithm
outperforms a classical (statically weighted) restricted Delaunay-refinement
technique for a number of three-dimensional benchmark problems.Comment: To appear at the 25th International Meshing Roundtabl
A Study of Different Modeling Choices For Simulating Platelets Within the Immersed Boundary Method
The Immersed Boundary (IB) method is a widely-used numerical methodology for
the simulation of fluid-structure interaction problems. The IB method utilizes
an Eulerian discretization for the fluid equations of motion while maintaining
a Lagrangian representation of structural objects. Operators are defined for
transmitting information (forces and velocities) between these two
representations. Most IB simulations represent their structures with
piecewise-linear approximations and utilize Hookean spring models to
approximate structural forces. Our specific motivation is the modeling of
platelets in hemodynamic flows. In this paper, we study two alternative
representations - radial basis functions (RBFs) and Fourier-based
(trigonometric polynomials and spherical harmonics) representations - for the
modeling of platelets in two and three dimensions within the IB framework, and
compare our results with the traditional piecewise-linear approximation
methodology. For different representative shapes, we examine the geometric
modeling errors (position and normal vectors), force computation errors, and
computational cost and provide an engineering trade-off strategy for when and
why one might select to employ these different representations.Comment: 33 pages, 17 figures, Accepted (in press) by APNU
Optimized normal and distance matching for heterogeneous object modeling
This paper presents a new optimization methodology of material blending for heterogeneous object modeling by matching the material governing features for designing a heterogeneous object. The proposed method establishes point-to-point correspondence represented by a set of connecting lines between two material directrices. To blend the material features between the directrices, a heuristic optimization method developed with the objective is to maximize the sum of the inner products of the unit normals at the end points of the connecting lines and minimize the sum of the lengths of connecting lines. The geometric features with material information are matched to generate non-self-intersecting and non-twisted connecting surfaces. By subdividing the connecting lines into equal number of segments, a series of intermediate piecewise curves are generated to represent the material metamorphosis between the governing material features. Alternatively, a dynamic programming approach developed in our earlier work is presented for comparison purposes. Result and computational efficiency of the proposed heuristic method is also compared with earlier techniques in the literature. Computer interface implementation and illustrative examples are also presented in this paper
Finite element methods for surface PDEs
In this article we consider finite element methods for approximating the solution of partial differential equations on surfaces. We focus on surface finite elements on triangulated surfaces, implicit surface methods using level set descriptions of the surface, unfitted finite element methods and diffuse interface methods. In order to formulate the methods we present the necessary geometric analysis and, in the context of evolving surfaces, the necessary transport formulae. A wide variety of equations and applications are covered. Some ideas of the numerical analysis are presented along with illustrative numerical examples
Efficient spatial modelling using the SPDE approach with bivariate splines
Gaussian fields (GFs) are frequently used in spatial statistics for their
versatility. The associated computational cost can be a bottleneck, especially
in realistic applications. It has been shown that computational efficiency can
be gained by doing the computations using Gaussian Markov random fields (GMRFs)
as the GFs can be seen as weak solutions to corresponding stochastic partial
differential equations (SPDEs) using piecewise linear finite elements. We
introduce a new class of representations of GFs with bivariate splines instead
of finite elements. This allows an easier implementation of piecewise
polynomial representations of various degrees. It leads to GMRFs that can be
inferred efficiently and can be easily extended to non-stationary fields. The
solutions approximated with higher order bivariate splines converge faster,
hence the computational cost can be alleviated. Numerical simulations using
both real and simulated data also demonstrate that our framework increases the
flexibility and efficiency.Comment: 26 pages, 7 figures and 3 table
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