| 1. |
- Alnæs, M.S., et al.
(författare)
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Unified framework for finite element assembly
- 2009
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Ingår i: International Journal of Computational Science and Engineering. - 1742-7185 .- 1742-7193. ; 4:4, s. 231-244
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Tidskriftsartikel (refereegranskat)abstract
- At the heart of any finite element simulation is the assembly of matrices and vectors from discrete variational forms. We propose a general interface between problem-specific and general-purpose components of finite element programs. This interface is called Unified Form-assembly Code (UFC). A wide range of finite element problems is covered, including mixed finite elements and discontinuous Galerkin methods. We discuss how the UFC interface enables implementations of variational form evaluation to be independent of mesh and linear algebra components. UFC does not depend on any external libraries, and is released into the public domain. Copyright © 2009, Inderscience Publishers.
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| 2. |
- Jansson, Johan, et al.
(författare)
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Algorithms and Data Structures for Multi-Adaptive Time-Stepping
- 2008
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Ingår i: ACM Transactions on Mathematical Software. - : Association for Computing Machinery (ACM). - 0098-3500 .- 1557-7295. ; 35:3
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Tidskriftsartikel (refereegranskat)abstract
- Multi-adaptive Galerkin methods are extensions of the standard continuous and discontinuous Galerkin methods for the numerical solution of initial value problems for ordinary or partial differential equations. In particular, the multi-adaptive methods allow individual and adaptive time steps to be used for different components or in different regions of space. We present algorithms for efficient multi-adaptive time-stepping, including the recursive construction of time slabs and adaptive time step selection. We also present data structures for efficient storage and interpolation of the multi-adaptive solution. The efficiency of the proposed algorithms and data structures is demonstrated for a series of benchmark problems.
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| 3. |
- Jansson, Johan, 1978, et al.
(författare)
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Computational modeling of dynamical systems
- 2005
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Ingår i: Mathematical Models and Methods in Applied Sciences. - 0218-2025. ; 15:3, s. 471-481
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Tidskriftsartikel (refereegranskat)abstract
- In this short note, we discuss the basic approach to computational modeling of dynamical systems. If a dynamical system contains multiple time scales, ranging from very fast to slow, computational solution of the dynamical system can be very costly. By resolving the fast time scales in a short time simulation, a model for the effect of the small time scale variation on large time scales can be determined, making solution possible on a long time interval. This process of computational modeling can be completely automated. Two examples are presented, including a simple model problem oscillating at a time scale of 10-9 computed over the time interval [0, 100], and a lattice consisting of large and small point masses. © World Scientific Publishing Company.
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| 4. |
- Kirby, R.C., et al.
(författare)
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A compiler for variational forms
- 2006
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Ingår i: ACM Transactions on Mathematical Software. - : Association for Computing Machinery (ACM). - 0098-3500 .- 0098-3500 .- 1557-7295. ; 32:3, s. 417-444
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Tidskriftsartikel (refereegranskat)abstract
- As a key step towards a complete automation of the finite element method, we present a new algorithm for automatic and efficient evaluation of multilinear variational forms. The algorithm has been implemented in the form of a compiler, the FEniCS Form Compiler (FFC). We present benchmark results for a series of standard variational forms, including the incompressible Navier-Stokes equations and linear elasticity. The speedup compared to the standard quadrature-based approach is impressive; in some cases the speedup is as large as a factor of 1000. © 2006 ACM.
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| 5. |
- Kirby, R.C., et al.
(författare)
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Benchmarking domain-specific compiler optimizations for variational forms
- 2008
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Ingår i: ACM Transactions on Mathematical Software. - : Association for Computing Machinery (ACM). - 0098-3500 .- 1557-7295. ; 35:2
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Tidskriftsartikel (refereegranskat)abstract
- We examine the effect of using complexity-reducing relations [Kirby et al. 2006] to generate optimized code for the evaluation of finite-element 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). © 2008 ACM.
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| 6. |
- Kirby, R.C., et al.
(författare)
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Efficient compilation of a class of variational forms
- 2007
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Ingår i: ACM Transactions on Mathematical Software. - : Association for Computing Machinery (ACM). - 0098-3500 .- 1557-7295. ; 33:3
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Tidskriftsartikel (refereegranskat)abstract
- We investigate the compilation of general multilinear variational forms over affines simplices and prove a representation theorem for the representation of the element tensor (element stiffness matrix) as the contraction of a constant reference tensor and a geometry tensor that accounts for geometry and variable coefficients. Based on this representation theorem, we design an algorithm for efficient pretabulation of the reference tensor. The new algorithm has been implemented in the FEniCS Form Compiler (FFC) and improves on a previous loop-based implementation by several orders of magnitude, thus shortening compile-times and development cycles for users of FFC. © 2007 ACM.
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| 7. |
- Kirby, R.C., et al.
(författare)
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Optimizing the evaluation of finite element matrices
- 2006
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Ingår i: SIAM Journal on Scientific Computing. - : Society for Industrial & Applied Mathematics (SIAM). - 1064-8275 .- 1095-7197. ; 27:3, s. 741-758
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Tidskriftsartikel (refereegranskat)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 Navier-Stokes operators. 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 multiply-add pair per entry. Up to sixth degree, we can do it in less than about two pairs. Preliminary low-degree results for Poisson and Navier-Stokes operators in three dimensions are also promising. © 2005 Society for Industrial and Applied Mathematics.
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| 8. |
- Kirby, R.C., et al.
(författare)
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Topological optimization of the evaluation of finite element matrices
- 2006
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Ingår i: SIAM Journal on Scientific Computing. - : Society for Industrial & Applied Mathematics (SIAM). - 1064-8275 .- 1095-7197. ; 28:1, s. 224-240
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Tidskriftsartikel (refereegranskat)abstract
- We present a topological framework for finding low-flop algorithms for evaluating element stiffness matrices associated with multilinear forms for finite element methods posed over straight-sided affine domains. This framework relies on phrasing the computation on each element as the contraction of each collection of reference element tensors with an element-specific geometric tensor. We then present a new concept of complexity-reducing relations that serve as distance relations between these reference element tensors. This notion sets up a graph-theoretic context in which we may find an optimized algorithm by computing a minimum spanning tree. We present experimental results for some common multilinear forms showing significant reductions in operation count and also discuss some efficient algorithms for building the graph we use for the optimization. © 2006 Society for Industrial and Applied Mathematics.
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