Cofactor function

[termi-official]

Reminder for me to factor out the cofactor matrix computation

Tensors.jl/src/math_ops.jl

Lines 80 to 132 in 258540f

	"""
	inv(::SecondOrderTensor)
	Computes the inverse of a second order tensor.
	# Examples
	```jldoctest
	julia> A = rand(Tensor{2,3})
	3×3 Tensor{2, 3, Float64, 9}:
	0.590845 0.460085 0.200586
	0.766797 0.794026 0.298614
	0.566237 0.854147 0.246837
	julia> inv(A)
	3×3 Tensor{2, 3, Float64, 9}:
	19.7146 -19.2802 7.30384
	6.73809 -10.7687 7.55198
	-68.541 81.4917 -38.8361
	```
	"""
	@generated function Base.inv(t::Tensor{2, dim}) where {dim}
	Tt = get_base(t)
	idx(i,j) = compute_index(Tt, i, j)
	if dim == 1
	ex = :($Tt((dinv, )))
	elseif dim == 2
	ex = quote
	v = get_data(t)
	$Tt((v[$(idx(2,2))] * dinv, -v[$(idx(2,1))] * dinv,
	-v[$(idx(1,2))] * dinv, v[$(idx(1,1))] * dinv))
	end
	else # dim == 3
	ex = quote
	v = get_data(t)
	$Tt(((v[$(idx(2,2))]*v[$(idx(3,3))] - v[$(idx(2,3))]*v[$(idx(3,2))]) * dinv,
	-(v[$(idx(2,1))]*v[$(idx(3,3))] - v[$(idx(2,3))]*v[$(idx(3,1))]) * dinv,
	(v[$(idx(2,1))]*v[$(idx(3,2))] - v[$(idx(2,2))]*v[$(idx(3,1))]) * dinv,
	-(v[$(idx(1,2))]*v[$(idx(3,3))] - v[$(idx(1,3))]*v[$(idx(3,2))]) * dinv,
	(v[$(idx(1,1))]*v[$(idx(3,3))] - v[$(idx(1,3))]*v[$(idx(3,1))]) * dinv,
	-(v[$(idx(1,1))]*v[$(idx(3,2))] - v[$(idx(1,2))]*v[$(idx(3,1))]) * dinv,
	(v[$(idx(1,2))]*v[$(idx(2,3))] - v[$(idx(1,3))]*v[$(idx(2,2))]) * dinv,
	-(v[$(idx(1,1))]*v[$(idx(2,3))] - v[$(idx(1,3))]*v[$(idx(2,1))]) * dinv,
	(v[$(idx(1,1))]*v[$(idx(2,2))] - v[$(idx(1,2))]*v[$(idx(2,1))]) * dinv))
	end
	end
	return quote
	$(Expr(:meta, :inline))
	dinv = 1 / det(t)
	@inbounds return $ex
	end
	end

into a separate function together with its analytical gradient. We need this one frequently for integration over the reference domain to connect with the normal in the deformed domain (see Nansons formula). Needs to check for performance regressions.

Users will for now probably use the identity $cof(F) = det(F) F^{-T}$, which is arguably less efficient.