Tensor Parser

All Expression fields supports tensor variables and operators for tensors. If you, for example, want the cross product between two vectors, simply type

directly in the Expression field. The symbol for the cross product is among the standard mathematical symbols defined by the Unicode standard. The other special symbols used by expressions are the (inner) dot product, A · B, and the nabla operator, ∇A. Press Ctrl+Space to get a list of the supported operations that includes any special characters. The system font must support the special symbols to display them properly; otherwise, the expression might not look correct. It is always possible to copy-paste them from an editor that supports Unicode input or directly from a Unicode character map.

There are also some functions that you can use to perform tensor operations — for example, the transpose of a matrix or the inverse of a matrix.

Use this section in combination with the features described in the Physics Builder Tools chapter.

The following table lists the operator symbols and operations that the tensor parser supports.

Operation	Precedence	Example
Cross product	Same as multiply	or cross(a,b)
Inner dot product	Same as multiply	or dot(a,b)
Double dot product	Same as multiply	a : b
Gradient	Function	∇a or gradient(a)
Tangential gradient	Function	gradientT(a)
Divergence	Function	or divergence(a)
Curl	Function	or curl(a)
Tangential curl	Function	curlT(a)
Inverse of a matrix	Function	inverse(a)
Determinant of a matrix	Function	determinant(a)
Transpose of a matrix	Function	or transpose(a)
Normalize a vector	Function	normalize(a)
Norm of a vector	Function	norm(a)
Sum over last index	Function	sum(a)
Maximum element in a vector	Function	maxElem(a)
Minimum element in a vector	Function	minElem(a)
Fill with variable to size	Function	array(a,{2,2}) = {{a,a},{a,a}} array(a,"3x1") = {a,a,a}
Specify elements of vector	Variable name	{1,2,3,4,5}
Specify elements of matrix	Variable name	{{11,12},{21,22}}
Expand elements to list of arguments	Variable name	Let r = {x,y,z} f(r.1..n) becomes f(x,y,z) g(r.1..2) becomes g(x,y)
Force symmetry in matrix	Function	Let M be a matrix that is symmetric, M = {{u*u, u*v, u*w}, {v*u, v*v, v*w}, {w*u, w*v, w*w}}. Symmetry cannot be detected because v*u is different than u*v by string comparison. The symmetric operator forces symmetry: symmetric(M) = {{u*u, u*v, u*w} , {u*v, v*v, v*w} , {u*w, v*w, w*w}}
Zero out in-plane components of a spatial vector	Function	Let r = {r,phi,z} in 2D axial symmetry zeroInPlane(r) = {0,0,z}
Zero out out-of-plane components of a spatial vector	Function	Let r = {r,phi,z} in 2D axial symmetry zeroOutOfPlane(r) = {r,0,z}
Multiply by volume integration factor	Function	integrand(a+b) becomes: (a+b)*2*pi*r in 2D axial symmetry (a+b)*ie1.detInvT for an infinite element domain
Evaluate physical constants, global parameters, and units to numerical values. You can use this operator with conditional expressions to check if a parameter is larger than a certain value, for example.	Function	Let T0 be a global parameter set to 300K, evalConst(k_B_const*T0/e_const) becomes 0.025851997154882865 evalConst(1[µm]) becomes 1e-6 if the base unit is meter.
commaDerivative	Function	Let u be a vector-valued dependent variable with components u, v, and w and the coordinate names x, y, and z. Then commaDerivative(u) = {{ux,uy,uz}, {vx,vy,vz}, {wx,wy,wz}}

The double dot product is a summation over two indices:

Unfortunately, there are two definitions of the double dot product, and the above is referred to the Frobenius inner product or the colon product. The other definition has flipped order for the indices in the second factor

The former definition is used by the tensor parser.

The gradient operator can be suffixed with s, m, g, or M to specify in regard to which coordinate variables (spatial, material, geometry, or mesh frame, respectively) it should take its derivatives. Example ∇.m.u is the gradient of the variable u in the material frame.