# Unified framework for finite element assembly

@article{Alns2009UnifiedFF, title={Unified framework for finite element assembly}, author={Martin Sandve Aln{\ae}s and Anders Logg and Kent‐Andre Mardal and Ola Skavhaug and Hans Petter Langtangen}, journal={Int. J. Comput. Sci. Eng.}, year={2009}, volume={4}, pages={231-244} }

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…

## 79 Citations

### Efficient implementation of finite element methods on non-matching and overlapping meshes in 3D

- Computer Science
- 2012

This note addresses the main challenges of an implementation of a fully general code for automatic detection of mesh intersections and integration over cut cells and subsimplices and considers Nitsche's method, a canonical example of an overlapping mesh method.

### Efficient Implementation of Finite Element Methods on Nonmatching and Overlapping Meshes in Three Dimensions

- Computer ScienceSIAM J. Sci. Comput.
- 2013

This note addresses the main challenges of an implementation of a fully general code for automatic detection of mesh intersections and integration over cut cells and subsimplices and considers Nitsche's method, which is a canonical example of an overlapping mesh method.

### Unified Embedded Parallel Finite Element Computations via Software-Based Fr[e-acute]chet Differentiation

- Computer ScienceSIAM J. Sci. Comput.
- 2010

A mathematical framework is presented that unifies the discretization of PDEs with these additional analysis requirements and provides high-level programming abstractions for the automatic, efficient evaluation of finite variational forms together with the derived operators required by engineering analysis.

### A stabilized Nitsche overlapping mesh method for the Stokes problem

- MathematicsNumerische Mathematik
- 2014

A Nitsche-based formulation for a general class of stabilized finite element methods for the Stokes problem posed on a pair of overlapping, non-matching meshes is developed and it is proved that the method is stable, consistent, and optimally convergent.

### Efficient representation of computational meshes

- Computer ScienceInt. J. Comput. Sci. Eng.
- 2009

A straightforward and efficient storage scheme for mesh representations and efficient algorithms for computation of arbitrary incidence relations from a given initial and minimal set of incidence relations are discussed.

### An automated hierarchical eXtended finite element approach for multiphysics problems involving discontinuities

- Computer Science
- 2018

The hierarchical eXtended finite element method is implemented as toolbox miXFEM for the FEniCS framework and can be used to rapidly implement and simulate problems from different areas such as multiphase flow or other problems with evolving geometries.

### Iterative solution of a mixed finite element discretisation of an incompressible magnetohydrodynamics problem

- Computer Science
- 2014

A broad-scope code has been generated using the finite element software package FEniCS and the linear algebra software PETSc to develop and numerically test a large scale preconditioned finite element implementation of an incompressible magnetohydrodynamics model.

### Automated Modelling of Evolving Discontinuities

- Computer ScienceAlgorithms
- 2009

The automated approximation of solutions to differential equations which involve discontinuities across evolving surfaces is addressed by exploring the scope for employing automated code generation techniques for modelling discontinUities.

### Numerical Construction of Static Fluid Interfaces with the Embedding Formalism

- Mathematics
- 2014

This research project develops a mathematical and numerical framework for representing static fluid interfaces as embedded manifolds. A variational principle is developed for the embedding function…

### Optimal Constrained Interpolation in Mesh-Adaptive Finite Element Modeling

- Computer ScienceSIAM J. Sci. Comput.
- 2017

Extensions to mesh-to-mesh Galerkin $L^2$ projection are considered which are optimal and ensure exact conservation of key discrete properties, including preservation of both the L^2 norm and the integral, and preservation of the $L ^2$ norm and weak incompressibility.

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From the Publisher:
This book is particularly useful to graduate students, researchers, and engineers using finite element methods. The reader should have knowledge of analysis and functional…