| 1. |
- Alnaes, Martin S., et al.
(författare)
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UFC: a Finite Element Code Generation Interface
- 2012
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Ingår i: Automated Solution of Differential Equations by the Finite Element Method. Anders Logg, Kent-Andre Mardal, Garth Wells (Eds.). - Heidelberg : Springer. - 9783642230981 ; , s. 283-302
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Bokkapitel (övrigt vetenskapligt/konstnärligt)abstract
- A central component of FEniCS is the UFC interface (Unified Form-assembly Code). UFC is an 8703 interface between problem-specific and general-purpose components of finite element programs. In 8704 particular, the UFC interface defines the structure and signature of the code that is generated by 8705 the form compilers FFC and SFC for DOLFIN. The UFC interface applies to a wide range of finite 8706 element problems (including mixed finite elements and discontinuous Galerkin methods) and may be 8707 used with libraries that differ widely in their design.
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| 2. |
- Alnaes, Martin, et al.
(författare)
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Unified Form Language: A Domain-Specific Language for Weak Formulations of Partial Differential Equations
- 2014
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Ingår i: ACM Transactions on Mathematical Software. - 0098-3500. ; 40:2, s. artikel nr 9-
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Tidskriftsartikel (refereegranskat)abstract
- We present the Unified Form Language (UFL), which is a domain-specific language for representing weak formulations of partial differential equations with a view to numerical approximation. Features of UFL include support for variational forms and functionals, automatic differentiation of forms and expressions, arbitrary function space hierarchies for multifield problems, general differential operators and flexible tensor algebra. With these features, UFL has been used to effortlessly express finite element methods for complex systems of partial differential equations in near-mathematical notation, resulting in compact, intuitive and readable programs. We present in this work the language and its construction. An implementation of UFL is freely available as an open-source software library. The library generates abstract syntax tree representations of variational problems, which are used by other software libraries to generate concrete low-level implementations. Some application examples are presented and libraries that support UFL are highlighted.
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| 3. |
- Arnold, Douglas N., et al.
(författare)
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Periodic Table of the Finite Elements
- 2014
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Ingår i: SIAM News. - 0036-1437. ; 47:9
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Tidskriftsartikel (refereegranskat)abstract
- The finite element method is one of the most powerful and widely applicable techniques for the numerical solution of partial differential equations and, therefore, for the simulation of the physical world. First proposed by engineers in the 1950s as a practical numerical method for predicting the deflection and stress of structural components of aircraft, the method has since been continuously extended and refined. It is now used in almost all application areas modeled by PDEs: solid and fluid dynamics, electromagnetics, biophysics, and even finance, to name just a few. Much as the chemical elements can be arranged in a periodic table based on their electron structure and recurring chemical properties, a broad assortment of finite elements can be arranged in a table that clarifies their properties and relationships. This arrangement, which is based on expression of the finite element function spaces in the language of differential forms, is one of the major outcomes of the theory known as finite element exterior calculus, or FEEC. Just as the arrangement of the chemical elements in a periodic table led to the discovery of new elements, the periodic table of finite elements has not only clarified existing elements but also highlighted holes in our knowledge and led to new families of finite elements suited for certain purposes.
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| 4. |
- Balaban, Gabriel, et al.
(författare)
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A Newton Method for Fluid-Structure Interaction Using Full Jacobians Based on Automatic Form Differentiation
- 2012
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Ingår i: 6th European Congress on Computational Methods in Applied Sciences and Engineering, ECCOMAS 2012; Vienna; Austria; 10 September 2012 through 14 September 2012. - 9783950353709 ; , s. 3434-3447
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Konferensbidrag (refereegranskat)abstract
- The study of fluid-structure interaction (FSI) problems is becoming increasingly important both as part of design/engineering and in the modeling of biomedical processes. Examples include the design of new fighter aircraft, the study of the dynamics of heart valves, and the design of prosthetic heart valves. FSI problems are highly coupled and highly nonlinear problems which are challenging to solve. Furthermore, the solution of the discretized system of nonlinear equations is particularly challenging in cases where the solid and the fluid have densities of similar size; this is typically the case for the simulation of biomedical processes involving the deformation of tissue. In such cases, a simple fixed point iteration, in which the solution from a fluid solver is used to impose Neumann boundary conditions for a structure (elasticity), followed by an update of the fluid domain based on the structure solution (via the solution of an auxiliary problem for the update of the fluid mesh), may fail to converge. Instead, a more coupled approach such as a Newton or quasi-Newton method must be employed. In this note, we study the use of Newton's method to solve the fully coupled FSI problem. Typically, a Lagrangian formulation is used to describe the solid; that is, the solid equations are solved on a fixed reference domain (the initial configuration), while an ALE (Arbitrary Lagrangian-Eulerian) formulation is used to describe the fluid. This means that the fluid domain is changing throughout the simulation of a time-dependent problem. The differentiation of the FSI problem, which is required to formulate Newton's method, therefore involves a differentiation with respect to the changing domain of the fluid problem. Such shape differentiation can indeed be used to derive the full Jacobian of the FSI problem; see Fernández and Moubachir [3]. We here study an alternative approach based on mapping the fluid problem back to the initial configuration of the fluid domain. This alternative is advantageous since it allows the use of straightforward differentiation on a fixed domain. This also allows the use of existing tools for automatic differentiation of finite element variational forms such as those developed as part of the FEniCS Project [5-7]. The FEniCS form language UFL [1] is a domain-specific language for finite element variational forms which allows the FSI problem to be expressed in a language close to the mathematical notation. Forms may be differentiated automatically, and automatically assembled into matrices and vectors. The methodology is here applied to the fully nonlinear time-dependent FSI problem modeled by the incompressible Navier-Stokes equations and the St. Venant-Kirchoff nonlinear hyperelastic model.
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| 5. |
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| 6. |
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| 7. |
- Kirby, Robert C., et al.
(författare)
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Common and Unusual Finite Elements
- 2012
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Ingår i: Automated Solution of Differential Equations by the Finite Element Method. Anders Logg, Kent-Andre Mardal, Garth Wells (Eds.). - Heidelberg : Springer. - 9783642230981 ; , s. 95-119
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Bokkapitel (övrigt vetenskapligt/konstnärligt)abstract
- This chapter provides a glimpse of the considerable range of finite elements in the literature. Many of the elements presented here are implemented as part of the FEniCS project already; some are future work. The universe of finite elements extends far beyond what we consider here. In particular, we consider only simplicial, polynomial-based elements. We thus bypass elements defined on quadrilaterals and hexahedra, composite and macro-element techniques, as well as XFEM-type methods. Even among polynomial-based elements on simplices, the list of elements can be extended. Nonetheless, this chapter presents a comprehensive collection of some the most common, and some more unusual, finite elements.
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| 8. |
- Kirby, Robert C., et al.
(författare)
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Discrete Optimization of Finite Element Matrix Evaluation
- 2012
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Ingår i: Automated Solution of Differential Equations by the Finite Element Method. Anders Logg, Kent-Andre Mardal, Garth Wells (Eds.). - Heidelberg : Springer. - 9783642230981 ; , s. 163-169
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Bokkapitel (övrigt vetenskapligt/konstnärligt)abstract
- The tensor contraction structure for the computation of the element tensor AT obtained in Chapter 8,enables not only the construction of a compiler for variational forms,but an optimizing compiler.For typical variational forms,the reference tensor A0 has significant structure that allows the element tensor AT to be computed on an arbitrary cell T at a lower computational cost.
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| 9. |
- Kirby, Robert C., et al.
(författare)
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FErari: an Optimizing Compiler for Variational Forms
- 2012
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Ingår i: Automated Solution of Differential Equations by the Finite Element Method. Anders Logg, Kent-Andre Mardal, Garth Wells (Eds.). - Heidelberg : Springer. - 9783642230981 ; , s. 239-246
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Bokkapitel (övrigt vetenskapligt/konstnärligt)abstract
- In Chapter 8, we presented a framework for efficient evaluation of multilinear forms based on 7311 expressing the multilinear form as a special tensor contraction. This allows generation of efficient 7312 low-level code for assembly of a range of multilinear forms.
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| 10. |
- Kirby, Robert C., et al.
(författare)
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Finite Element Variational Forms
- 2012
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Ingår i: Automated Solution of Differential Equations by the Finite Element Method. Anders Logg, Kent-Andre Mardal, Garth Wells (Eds.). - Heidelberg : Springer. - 9783642230981 ; , s. 133-140
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Bokkapitel (övrigt vetenskapligt/konstnärligt)abstract
- Much of the FEniCS software is devoted to the formulation of variational forms (UFL), the discretization of variational forms (FIAT, FFC, SyFi) and the assembly of the corresponding discrete operators (UFC, DOLFIN). This chapter summarizes the notation for variational forms used throughout FEniCS.
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