1. 
 Alnæs, Martin S., et al.
(författare)

The FEniCS Project Version 1.5
 2015

Ingår i: Archive of Numerical Software.  21978263 . 21978263. ; 3:100, s. 923

Tidskriftsartikel (refereegranskat)abstract
 The FEniCS Project is a collaborative project for the development ofinnovative concepts and tools for automated scientific computing, with a particular focus on the solution of differential equations byfinite element methods. The FEniCS Projects software consists of a collection of interoperable software components, including DOLFIN,FFC, FIAT, Instant, UFC, UFL, and mshr. This note describes the newfeatures and changes introduced in the release of FEniCSversion 1.5.


2. 
 Alnaes, Martin S., et al.
(författare)

UFC: a Finite Element Code Generation Interface
 2012

Ingår i: Automated Solution of Differential Equations by the Finite Element Method. Anders Logg, KentAndre Mardal, Garth Wells (Eds.).  Heidelberg : Springer.  9783642230981 ; s. 283302

Bokkapitel (övrigt vetenskapligt)abstract
 A central component of FEniCS is the UFC interface (Unified Formassembly Code). UFC is an 8703 interface between problemspecific and generalpurpose 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.


3. 
 Alnæs, M.S., et al.
(författare)

Unified framework for finite element assembly
 2009

Ingår i: International Journal of Computational Science and Engineering.  17427185 . 17427193. ; 4:4, s. 231244

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 problemspecific and generalpurpose components of finite element programs. This interface is called Unified Formassembly 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.


4. 
 Alnaes, Martin, et al.
(författare)

Unified Form Language: A DomainSpecific Language for Weak Formulations of Partial Differential Equations
 2014

Ingår i: ACM Transactions on Mathematical Software.  00983500. ; 40:2, s. artikel nr 9

Tidskriftsartikel (refereegranskat)abstract
 We present the Unified Form Language (UFL), which is a domainspecific 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 nearmathematical 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 opensource software library. The library generates abstract syntax tree representations of variational problems, which are used by other software libraries to generate concrete lowlevel implementations. Some application examples are presented and libraries that support UFL are highlighted.


5. 
 Ames, E., et al.
(författare)

Cosmic string and black hole limits of toroidal Vlasov bodies in general relativity
 2019

Ingår i: Physical Review D.  24700010. ; 99:2

Tidskriftsartikel (refereegranskat)abstract
 We numerically investigate limits of a twoparameter family of stationary solutions to the EinsteinVlasov system. The solutions are toroidal and have nonvanishing angular momentum. As the parameters are tuned to more relativistic solutions (measured e.g., by an increasing redshift) we provide evidence for a sequence of solutions which approaches the extreme Kerr black hole family. Solutions with angular momentum larger than the square of the mass are also investigated, and in the relativistic limit the nearfield geometry of such solutions is observed to become locally rotationally symmetric about the matter density. The existence of a deficit angle in these regions is investigated.


6. 
 Ames, Ellery, et al.
(författare)

On axisymmetric and stationary solutions of the selfgravitating Vlasov system
 2016

Ingår i: Classical and Quantum Gravity.  02649381. ; 33:15

Tidskriftsartikel (refereegranskat)abstract
 Axisymmetric and stationary solutions are constructed to the EinsteinVlasov and VlasovPoisson systems. These solutions are constructed numerically, using finite element methods and a fixedpoint iteration in which the total mass is fixed at each step. A variety of axisymmetric stationary solutions are exhibited, including solutions with toroidal, disklike, spindlelike, and composite spatial density configurations, as are solutions with nonvanishing net angular momentum. In the case of toroidal solutions, we show for the first time, solutions of the EinsteinVlasov system which contain ergoregions.


7. 
 Arnold, Douglas N., et al.
(författare)

Periodic Table of the Finite Elements
 2014

Ingår i: SIAM News.  00361437. ; 47:9

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.


8. 
 Balaban, Gabriel, et al.
(författare)

A Newton Method for FluidStructure Interaction Using Full Jacobians Based on Automatic Form Differentiation
 2012

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. 34343447

Konferensbidrag (refereegranskat)abstract
 The study of fluidstructure 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 quasiNewton 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 LagrangianEulerian) formulation is used to describe the fluid. This means that the fluid domain is changing throughout the simulation of a timedependent 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 [57]. The FEniCS form language UFL [1] is a domainspecific 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 timedependent FSI problem modeled by the incompressible NavierStokes equations and the St. VenantKirchoff nonlinear hyperelastic model.


9. 


10. 
 Eriksson, Kenneth, 1952, et al.
(författare)

Explicit timestepping for stiff ODEs
 2003

Ingår i: SIAM Journal on Scientific Computing.  10648275. ; 25:4, s. 11421157

Tidskriftsartikel (refereegranskat)abstract
 We present a new strategy for solving stiff ODEs with explicit methods. By adaptively taking a small number of stabilizing small explicit time steps when necessary, a stiff ODE system can be stabilized enough to allow for time steps much larger than what is indicated by classical stability analysis. For many stiff problems the cost of the stabilizing small time steps is small, so the improvement is large. We illustrate the technique on a number of wellknown stiff test problems.

