Algebra Basics

This page contains a basic information about mathematical operators and algebric operations that can be perfomed in Swan.

Differential Operators

Differential operators are currently defined for Lagrangian Functions.

Operation	Expression	Notes
Partial Derivative	Partial(u, i)
Gradient	Grad(u)
Symmetric Gradient	SymGrad(u)
Divergence	Divergence(u)
Laplacian	u.computeLaplacianFun(xV)	xV are the Gauss points
Curl	u.computeCurl()	Only for 2D
Deviatoric	Deviatoric(A)
Volumetric Strain	VolumetricStrain(u)
Deviatoric Strain	DeviatoricStrain(u)
Deviatoric Elastic Energy Density	DeviatoricElasticEnergyDensity(u,mu)
Volumetric Elastic Energy Density	VolumetricElasticEnergyDensity(u,mu)

Algebric Operations

In this section we describe operations can be defined for DomainFunctions or LagrangianFunction.

Operation	Expresion	Notes
Pointwise Multiplication	times(a,A)	Scalar-Scalar, Scalar-Vector or Scalar-Tensor multiplications
Dot Product	DP(A,B)	Single Index Contraction for Tensors
Double Dot Product	DDP(A,B)	Double Index Contraction for Tensors

Integral Operations

Function	Expression	Notes
Mean	Mean(f,quad)	Mean value of f over the mesh with quadrature order quad
Integral	Integrate(f)	Integral of f over the mesh

Nonlinear Functions

Function	Expression
Logarithm	log(u)
Exponential	exp(u)
Square Root	sqrt(u)
Heaviside	Heaviside(x)

Norms and Inner Products

Operation	Definition	Expression
L2 Inner Product	⟨f, g⟩ = ∫ f ⋅ g	ScalarProduct(f,g,'L2')
L2 Norm	‖f‖ = sqrt(⟨a, a⟩)	Norm(f,'L2')
H1 Inner Product	⟨f, g⟩ = ∫f ⋅ g + ε∇f ⋅ ∇g	ScalarProduct(f,g,'H1',epsilon)
H1 Norm	‖f‖ = sqrt(⟨f, f⟩)	Norm(f,'H1', epsilon)