Vector Laplacian

[mhelel]

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Question

I want to implement the vector laplacian $\nabla\dot(\nabla\vec{u}) = \nabla^2 \vec{u}$ in its strong form inside a Material that just uses a single vector variable $\vec{u}$ instead of three components. Therefore I need the second spatial derivatives of $\vec{u}$.

I know that if I would just have the components as coupled variables, I could just use the coupledSecond option and If I would use AD, which I don't, it seems that there is a adCoupledVectorSecond option. But since both cases are not applicable I was wondering how to solve this problem. In another discussion one was referred to https://mooseframework.inl.gov/docs/doxygen/moose/classMooseVariableFE.html#a6084c139e379961bfcc3f5ed97d181f4 but I was not able to figure out, how this should help. If this is the right direction, I would appreciate an hint, how to implement this directly.

[GiudGiud]

Hello

We typically integrate this term by parts instead when it shows up in the residual of the weak form. So it has not been implemented for vector varaibles, only for regular fields. Also the "adCoupledVectorSecond" option is just returning an error I think. Otherwise you could have just done " ADNumber.value() " to get the non-AD value

But if you really need coupledSecond for vector variables, this is going to require some development.
I started on a branch here:
https://github.com/GiudGiud/moose/tree/PR_second_vec

but it needs a change in libmesh, implementing slice operations for order N tensors. So it will be delayed
The libMesh PR is there:
libMesh/libmesh#4343
It's not finished. Feel free to build on it

[lindsayad]

We work around this for the AD implementation in this way:

moose/modules/navier_stokes/include/materials/INSADTauMaterial.h

Lines 257 to 277 in c481cda

	// libMesh does not yet have the capability for computing second order spatial derivatives of
	// vector bases. Lacking that capability, we can compute the second order spatial derivatives "by
	// hand" using the scalar field version of the vector basis, e.g. LAGRANGE instead of
	// LAGRANGE_VEC. Adding this implementation allows us to be fully consistent with results from a
	// scalar velocity field component implementation of Navier-Stokes
	const auto & vel_dof_indices = _velocity_var->dofIndices();
	for (const auto i : index_range(vel_dof_indices))
	{
	// This may not work if the element and spatial dimensions are different
	mooseAssert(_current_elem->dim() == _mesh.dimension(),
	"Below logic only applicable if element and mesh dimension are the same");
	const auto dimensional_component = i % _mesh.dimension();
	auto & d2vel = get_d2(dimensional_component);
	const auto dof_index = vel_dof_indices[i];
	ADReal dof_value = (*_velocity_var->sys().currentSolution())(dof_index);
	if (doVelocityDerivatives())
	dof_value.derivatives().insert(dof_index) = 1;
	const auto scalar_i_component = i / _mesh.dimension();
	for (const auto qp : make_range(_qrule->n_points()))
	d2vel[qp] += dof_value * _scalar_lagrange_fe->get_d2phi()[scalar_i_component][qp];
	}

But you should also see my response to your other discussion #32015