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14
Geometric optimization of the evaluation of finite element matrices
 SIAM J. Sci. Comput
"... Abstract. Assembling stiffness matrices represents a significant cost in many finite element computations. We address the question of optimizing the evaluation of these matrices. By finding redundant computations, we are able to significantly reduce the cost of building local stiffness matrices for ..."
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Abstract. Assembling stiffness matrices represents a significant cost in many finite element computations. We address the question of optimizing the evaluation of these matrices. By finding redundant computations, we are able to significantly reduce the cost of building local stiffness matrices for the Laplace operator and for the trilinear form for NavierStokes. For the Laplace operator in two space dimensions, we have developed a heuristic graph algorithm that searches for such redundancies and generates code for computing the local stiffness matrices. Up to cubics, we are able to build the stiffness matrix on any triangle in less than one multiplyadd pair per entry. Up to sixth degree, we can do it in less than about two. Preliminary lowdegree results for Poisson and NavierStokes operators in three dimensions are also promising.
DOLFIN: Automated finite element computing
, 2009
"... We describe here a library aimed at automating the solution of partial differential equations using the finite element method. By employing novel techniques for automated code generation, the library combines a high level of expressiveness with efficient computation. Finite element variational forms ..."
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Cited by 19 (0 self)
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We describe here a library aimed at automating the solution of partial differential equations using the finite element method. By employing novel techniques for automated code generation, the library combines a high level of expressiveness with efficient computation. Finite element variational forms may be expressed in near mathematical notation, from which lowlevel code is automatically generated, compiled and seamlessly integrated with efficient implementations of computational meshes and highperformance linear algebra. Easytouse objectoriented interfaces to the library are provided in the form of a C++ library and a Python module. This paper discusses the mathematical abstractions and methods used in the design of the library and its implementation. A number of examples are presented to demonstrate the use of the library in application code.
Benchmarking domainspecific compiler optimizations for variational forms
 CODEN ACMSCU. ISSN 00983500 (print), 15577295 (electronic). QuintanaOrti:2009:ULF
, 2009
"... We examine the effect of using complexityreducing relations [Kirby et al. 2006] to generate optimized code for the evaluation of finiteelement variational forms. The optimizations are implemented in a prototype code named FErari, which has been integrated as an optimizing backend to the FEniCS f ..."
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Cited by 5 (2 self)
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We examine the effect of using complexityreducing relations [Kirby et al. 2006] to generate optimized code for the evaluation of finiteelement variational forms. The optimizations are implemented in a prototype code named FErari, which has been integrated as an optimizing backend to the FEniCS form compiler, FFC [Kirby and Logg 2006; 2007]. In some cases, FErari provides very little speedup, while in other cases we obtain reduced local operation counts by a factor of as much as 7.9 and speedups for the assembly of the global sparse matrix by as much as a factor of 2.8 (see Figure 9).
Multiscale modeling of the acoustic properties of lung parenchyma
 ESAIM: Proc
"... Abstract. Lung parenchyma is a foamlike material consisting of millions of alveoli. Sound transmission through parenchyma plays an important role in the noninvasive diagnosis of many lung diseases. We model the parenchyma as a porous solid with air filled pores and consider the Biot equations as ..."
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Abstract. Lung parenchyma is a foamlike material consisting of millions of alveoli. Sound transmission through parenchyma plays an important role in the noninvasive diagnosis of many lung diseases. We model the parenchyma as a porous solid with air filled pores and consider the Biot equations as a model for its acoustic properties. The Biot equations govern smallamplitude wave propagation in fluidsaturated porous solids, and include the effects of relative motion between the fluid and the solid frame. The Biot equations can be derived from a microstructure model of the porous material, and the material parameters in the equations can be obtained from the solution of two independent microstructure problems, a fluidcell problem (governed by the unsteady Stokes equations) and a solidcell problem. We review the homogenization approach for media with periodic microstructure, and solve a fluidcell problem numerically for an idealized twodimensional microscale geometry for a wide range of frequencies. We also discuss sound speeds in lung tissue. Résumé.... 1.
On the Efficiency of Symbolic Computations Combined with Code Generation for Finite Element Methods
"... Efficient and easy implementation of variational forms for finite element discretization can be accomplished with metaprogramming. Using a highlevel language like Python and symbolic mathematics makes an abstract problem definition possible, but the use of a lowlevel compiled language is vital fo ..."
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Efficient and easy implementation of variational forms for finite element discretization can be accomplished with metaprogramming. Using a highlevel language like Python and symbolic mathematics makes an abstract problem definition possible, but the use of a lowlevel compiled language is vital for runtime efficiency. By generating lowlevel C++ code based on symbolic expressions for the discrete weak form, it is possible to accomplish a high degree of abstraction in the problem definition while surpassing the runtime efficiency of traditional hand written C++ codes. We provide several examples where we demonstrate orders of magnitude in speedup.
Representations of finite element tensors via automated code generation
 ACM Transactions on Mathematical Software
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SIMPSON ET AL.: A MULTISCALE MODEL OF PARTIAL MELTS In a companion paper, equations for partially molten media
, 903
"... 20 were derived using twoscale homogenization theory. One advantage of homogenization is that material properties, such as permeability and viscosity, readily emerge. A caveat is that the dependence of these parameters upon the microstructure is not selfevident. In particular, one seeks to relate ..."
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20 were derived using twoscale homogenization theory. One advantage of homogenization is that material properties, such as permeability and viscosity, readily emerge. A caveat is that the dependence of these parameters upon the microstructure is not selfevident. In particular, one seeks to relate them to the porosity. In this paper, we numerically solve ensembles of the cell problems from which these quantities emerge. Using this data, we estimate relationships between the parameters and the porosity. In particular, the bulk viscosity appears to be inversely proportional to the porosity. Finally, we synthesize these numerical estimates with the models. Our hybrid numerical–
5 6
, 903
"... were derived using twoscale homogenization theory. This approach begins with a grain scale description and then coarsens it through multiple scale expansions into a macroscopic model. One advantage of homogenization is that effective material properties, such as permeability and the shear and bulk ..."
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were derived using twoscale homogenization theory. This approach begins with a grain scale description and then coarsens it through multiple scale expansions into a macroscopic model. One advantage of homogenization is that effective material properties, such as permeability and the shear and bulk viscosity of the twophase medium, are characterized by cell problems, boundary value problems posed on a representative microstructural cell. The solutions of these problems can be averaged to obtain macroscopic parameters that are consistent with a given microstructure. This is particularly important for estimating the “compaction length ” which depends on the product of permeability and bulk viscosity and is the intrinsic length scale for viscously deformable twophase flow. In this paper, we numerically solve ensembles of cell problems for several geometries. We begin with simple intersecting tubes as this is a one parameter family of problems with well known results for permeability. Using this data, we estimate relationships between the porosity and all of the effective
Unified Framework for . . .
"... Over the last fifty years, the finite element method has emerged as a successful methodology for solving a wide range of partial differential equations. At the heart of any finite element simulation is the assembly of matrices and vectors from finite element variational forms. In this paper, we pre ..."
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Over the last fifty years, the finite element method has emerged as a successful methodology for solving a wide range of partial differential equations. At the heart of any finite element simulation is the assembly of matrices and vectors from finite element variational forms. In this paper, we present a general and unified framework for finite element assembly. Based on this framework, we propose a specific software interface (UFC) between problemspecific and generalpurpose components of finite element programs. The interface is general in the sense that it applies to a wide range of finite element problems (including mixed finite elements and discontinuous Galerkin methods) and may be used with libraries that differ widely in their design. The interface consists of a minimal set of abstract C++ classes and data transfer is via plain C arrays. We discuss how one may use the UFC interface to build a plugandplay system for finite element simulation where basic components such as computational meshes, linear algebra and, in particular, variational form evaluation may come from different libraries and be used interchangeably. We further discuss how the UFC interface is used to glue
Relevance Automation of Error Control with Application to Fluid–Structure Interaction in Biomedicine
"... Computer simulation is an important tool in many areas of science. Increasingly complex mathematical models are being solved in large computer simulations, complementing and sometimes replacing traditional experimental techniques as the main tool of scientific investigation. In any such computer sim ..."
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Computer simulation is an important tool in many areas of science. Increasingly complex mathematical models are being solved in large computer simulations, complementing and sometimes replacing traditional experimental techniques as the main tool of scientific investigation. In any such computer simulation, it is pivotal that the quality of the computed solution may be determined. However, the assessment of the quality of a computed solution is challenging, both mathematically and computationally. As a consequence, the quality of the solution must often be assessed manually by the scientist or engineer running the simulation. This is unreliable as well as timeconsuming, and it effectively prevents computer simulation from realizing its full potential as a standard tool in science and industry. Computer simulations often require large computational resources, in particular the simulation of complex biological processes studied in the proposed research. It is therefore of utmost importance that computational resources are used as efficiently as possible, to make new results readily available and to expand the realm of which processes may be simulated. We thus identify reliability and efficiency as two key challenges in computer simulation. These two challenges are addressed by error control. By (adaptive) error control, the resolution of the simulation is chosen such that the computed solution satisfies a given accuracy requirement with minimal work. Error control thus makes computer simulation accessible