Finite Elements

The Finite Element Method (FEM) is a widely used numerical technique, employed for the solution of partial differential equations (PDEs), arising in structural engineering analysis and more broadly in the field of continuum mechanics.

However, due to their generality and mathematically sound foundation, Finite Elements, are often employed in the implementation of other numerical schemes and for various computational operators, e.g. interpolation, integration, etc.

Mint provides basic support for Finite Elements that consists:

1. Lagrange Basis shape functions for commonly employed Cell Types

2. Corresponding Quadratures (under development)

3. Routines for forward/inverse Isoparametric Mapping, and

4. Infrastructure to facilitate adding shape functions for new Cell Types, as well as, to Add a New Finite Element Basis.

This functionality is collectively exposed to the application through the mint::FiniteElement class. Concrete examples illustrating the usage of the mint::FiniteElement class within an application code are provided in the Finite Elements tutorial section.

A Finite Element Basis consists of a family of shape functions corresponding to different Cell Types. Mint currently supports Lagrange isoparametric Finite Elements.

Lagrange Basis

The Lagrange basis consists of Cell Types whose shape functions are formed from products of the one-dimensional Lagrange polynomial. This section provides a summary of supported Lagrange Basis Cell Types, their associated shape functions, and summarize the process to Add a New Lagrange Element.

Note

The shape functions of all Lagrange Cells in Mint, follow the CGNS Numbering Conventions and are defined within a reference coordinate system, on the closed interval \(\hat{\xi} \in [0,1]\).

QUAD: Linear Quadrilateral

\begin{array}{r c l c l} N_0 &=& (1 - \xi) &\times& (1 - \eta) \\ N_1 &=& \xi &\times& (1 - \eta) \\ N_2 &=& \xi &\times& \eta \\ N_3 &=& (1 - \xi) &\times& \eta \\ \end{array}

QUAD9: Quadratic Quadrilateral

\begin{array}{r c l c l} N_0 &=& (\xi-1)( 2\xi -1) &\times& (\eta-1)(2\eta-1) \\ N_1 &=& \xi(2\xi-1) &\times& (\eta-1)(2\eta-1) \\ N_2 &=& \xi(2\xi-1) &\times& \eta(2\eta-1) \\ N_3 &=& (\xi-1)( 2\xi -1) &\times& \eta(2\eta-1) \\ \\ N_4 &=& 4\xi(1-\xi) &\times& (\eta-1)(2\eta-1) \\ N_5 &=& \xi(2\xi-1) &\times& 4\eta(1-\eta) \\ N_6 &=& 4\xi(1-\xi) &\times& \eta(2\eta-1) \\ N_7 &=& (\xi-1)( 2\xi -1) &\times& 4\eta(1-\eta) \\ \\ N_8 &=& 4\xi(1-\xi) &\times& 4\eta(1-\eta) \\ \end{array}

TRIANGLE: Linear Triangle

\begin{array}{r c l} N_0 & = & 1 - \xi - \eta \\ N_1 & = & \xi \\ N_2 & = & \eta \\ \end{array}

HEX: Linear Hexahedron

\begin{array}{r c l c l c l} N_0 &=& (1-\xi) &\times& (1-\eta) &\times& (1-\zeta) \\ N_1 &=& \xi &\times& (1-\eta) &\times& (1-\zeta) \\ N_2 &=& \xi &\times& \eta &\times& (1-\zeta) \\ N_3 &=& (1-\xi) &\times& \eta &\times& (1-\zeta) \\ \\ N_4 &=& (1-\xi) &\times& (1-\eta) &\times& \zeta \\ N_5 &=& \xi &\times& (1-\eta) &\times& \zeta \\ N_6 &=& \xi &\times& \eta &\times& \zeta \\ N_7 &=& (1-\xi) &\times& \eta &\times& \zeta \\ \end{array}

HEX27: Quadratic Hexahedron

\begin{array}{r c l c l c l} N_0 &=& (\xi-1)(2\xi-1) &\times& (\eta-1)(2\eta-1) &\times& (\zeta-1)(2\zeta-1) \\ N_1 &=& \xi(2\xi-1) &\times& (\eta-1)(2\eta-1) &\times& (\zeta-1)(2\zeta-1) \\ N_2 &=& \xi(2\xi-1) &\times& \eta(2\eta-1) &\times& (\zeta-1)(2\zeta-1) \\ N_3 &=& (\xi-1)(2\xi-1) &\times& \eta(2\eta-1) &\times& (\zeta-1)(2\zeta-1) \\ \\ N_4 &=& (\xi-1)(2\xi-1) &\times& (\eta-1)(2\eta-1) &\times& \zeta(2\zeta-1) \\ N_5 &=& \xi(2\xi-1) &\times& (\eta-1)(2\eta-1) &\times& \zeta(2\zeta-1) \\ N_6 &=& \xi(2\xi-1) &\times& \eta(2\eta-1) &\times& \zeta(2\zeta-1) \\ N_7 &=& (\xi-1)(2\xi-1) &\times& \eta(2\eta-1) &\times& \zeta(2\zeta-1) \\ \\ N_8 &=& 4\xi(1-\xi) &\times& (\eta-1)(2\eta-1) &\times& (\zeta-1)(2\zeta-1) \\ N_9 &=& \xi(2\xi-1) &\times& 4\eta(1-\eta) &\times& (\zeta-1)(2\zeta-1) \\ N_{10} &=& 4\xi(1-\xi) &\times& \eta(2\eta-1) &\times& (\zeta-1)(2\zeta-1) \\ N_{11} &=& (\xi-1)(2\xi-1) &\times& 4\eta(1-\eta) &\times& (\zeta-1)(2\zeta-1) \\ \\ N_{12} &=& 4\xi(1-\xi) &\times& (\eta-1)(2\eta-1) &\times& \zeta(2\zeta-1) \\ N_{13} &=& \xi(2\xi-1) &\times& 4\eta(1-\eta) &\times& \zeta(2\zeta-1) \\ N_{14} &=& 4\xi(1-\xi) &\times& \eta(2\eta-1) &\times& \zeta(2\zeta-1) \\ N_{15} &=& (\xi-1)(2\xi-1) &\times& 4\eta(1-\eta) &\times& \zeta(2\zeta-1) \\ \\ N_{16} &=& (\xi-1)(2\xi-1) &\times& (\eta-1)(2\eta-1) &\times& 4\zeta(1-\zeta) \\ N_{17} &=& \xi(2\xi-1) &\times& (\eta-1)(2\eta-1) &\times& 4\zeta(1-\zeta) \\ N_{18} &=& \xi(2\xi-1) &\times& \eta(2\eta-1) &\times& 4\zeta(1-\zeta) \\ N_{19} &=& (\xi-1)(2\xi-1) &\times& \eta(2\eta-1) &\times& 4\zeta(1-\zeta) \\ \\ N_{20} &=& (\xi-1)(2\xi-1) &\times& 4\eta(1-\eta) &\times& 4\zeta(1-\zeta) \\ N_{21} &=& \xi(2\xi-1) &\times& 4\eta(1-\eta) &\times& 4\zeta(1-\zeta) \\ N_{22} &=& 4\xi(1-\xi) &\times& (\eta-1)(2\eta-1) &\times& 4\zeta(1-\zeta) \\ N_{23} &=& 4\xi(1-\xi) &\times& \eta(2\eta-1) &\times& 4\zeta(1-\zeta) \\ N_{24} &=& 4\xi(1-\xi) &\times& 4\eta(1-\eta) &\times& (\zeta-1)(2\zeta-1) \\ N_{25} &=& 4\xi(1-\xi) &\times& 4\eta(1-\eta) &\times& \zeta(2\zeta-1) \\ \\ N_{26} &=& 4\xi(1-\xi) &\times& 4\eta(1-\eta) &\times& 4\zeta(1-\zeta) \\ \end{array}

PYRAMID: Linear Pyramid

\begin{array}{r c l c l c l} N_0 &=& (1-\xi) &\times& (1-\eta) &\times& (1-\zeta) \\ N_1 &=& \xi &\times& (1-\eta) &\times& (1-\zeta) \\ N_2 &=& \xi &\times& \eta &\times& (1-\zeta) \\ N_3 &=& (1-\xi) &\times& \eta &\times& (1-\zeta) \\ N_4 &=& \zeta \\ \end{array}

PRISM: Linear Prism/Wedge

\begin{array}{r c l c l c l} N_0 &=& (1-\xi) - \eta &\times& (1-\zeta) \\ N_1 &=& \xi &\times& (1-\zeta) \\ N_2 &=& \eta &\times& (1-\zeta) \\ N_3 &=& (1-\xi) - \eta &\times& \zeta \\ N_4 &=& \xi &\times& \zeta \\ N_5 &=& \eta &\times& \zeta \\ \end{array}

Add a New Lagrange Element

Warning

This section is under construction.

Isoparametric Mapping

Warning

This section is under construction.

Quadratures

Warning

Support for Quadratures in Mint is under development.

Add a New Finite Element Basis

Warning

This section is under construction.