Concepts inEfficient compilation of a class of variational forms
Differential form
In the mathematical fields of differential geometry and tensor calculus, differential forms are an approach to multivariable calculus that is independent of coordinates. Differential forms provide a unified approach to defining integrands over curves, surfaces, volumes, and higher dimensional manifolds. The modern notion of differential forms was pioneered by Élie Cartan. It has many applications, especially in geometry, topology and physics.
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Calculus of variations
Calculus of variations is a field of mathematics, or more specifically calculus, that deals with maximizing or minimizing functionals, which are mappings from a set of functions to the real numbers. Functionals are often expressed as definite integrals involving functions and their derivatives. The interest is in extremal functions that make the functional attain a maximum or minimum value ¿ or stationary functions ¿ those where the rate of change of the functional is zero.
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Compiler
A compiler is a computer program (or set of programs) that transforms source code written in a programming language (the source language) into another computer language (the target language, often having a binary form known as object code). The most common reason for wanting to transform source code is to create an executable program. The name "compiler" is primarily used for programs that translate source code from a high-level programming language to a lower level language.
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FEniCS Project
The FEniCS Project is a collection of free, open source, software components with the common goal to enable automated solution of differential equations. The components provide scientific computing tools for working with computational meshes, finite element variational formulations of ordinary and partial differential equations, and numerical linear algebra. The current stable version of the FEniCS Project is 1.0; released December 7 2011.
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Tensor contraction
In multilinear algebra, a tensor contraction is an operation on one or more tensors that arises from the natural pairing of a finite-dimensional vector space and its dual. In components, it is expressed as a sum of products of scalar components of the tensor(s) caused by applying the summation convention to a pair of dummy indices which are bound to each other in an expression.
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Tensor
whose columns are the forces acting on the,, and faces of the cube. ]] Tensors are geometric objects that describe linear relations between vectors, scalars, and other tensors. Elementary examples of such relations include the dot product, the cross product, and linear maps. Vectors and scalars themselves are also tensors. A tensor can be represented as a multi-dimensional array of numerical values.
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Stiffness matrix
For the stiffness tensor in solid mechanics, see Hooke's law#Matrix representation (stiffness tensor). In the finite element method and in analysis of spring systems, a stiffness matrix, K, is a symmetric positive-semidefinite matrix that generalizes the stiffness of Hooke's law to a matrix, describing the stiffness of between all of the degrees of freedom so that where F and x are the force and the displacement vectors, and is the system's total potential energy.
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