[StructuralMechanics] Add direct harmonic strategy (with imaginary displacements)

[juancamarotti]

Summary

This PR adds a new direct harmonic analysis strategy to the StructuralMechanicsApplication, together with the required infrastructure to prescribe imaginary harmonic loads and a dedicated test.

Main changes

• Added a new DirectHarmonicAnalysisStrategy
• Added a new direct harmonic analysis test
• Added an ImaginaryPointLoadCondition to prescribe imaginary load contributions separately
• Introduced the use of two different load SubModelParts:
  • one for the real harmonic loads
  • one for the imaginary harmonic loads

Mathematical meaning of the direct harmonic strategy

The direct harmonic strategy solves the structural response directly in the frequency domain by assuming a harmonic ansatz for the unknown displacement field:

$\mathbf{u}(t) = \Re\left( \hat{\mathbf{u}} e^{i \omega t} \right)$

where:

• $\hat{\mathbf{u}}$ is the complex displacement amplitude
• $\omega$ is the excitation circular frequency
• $\Re(\cdot)$ denotes the real part

Starting from the dynamic equilibrium equation

$\mathbf{M}\ddot{\mathbf{u}} + \mathbf{C}\dot{\mathbf{u}} + \mathbf{K}\mathbf{u} = \mathbf{f}(t)$

and assuming a harmonic load

$\mathbf{f}(t) = \Re\left( \hat{\mathbf{f}} e^{i \omega t} \right)$

the problem is transformed into the complex algebraic system

$\left( \mathbf{K} - \omega^2 \mathbf{M} + i \omega \mathbf{C} \right)\hat{\mathbf{u}} = f_{r} + i f_i$

This is what is referred to here as the direct harmonic strategy:
instead of projecting the response onto a modal basis, the full frequency-domain system is assembled and solved directly for the complex displacement vector $\hat{\mathbf{u}}$.

Details

Direct harmonic strategy

A new DirectHarmonicAnalysisStrategy was implemented to solve the harmonic response directly in the frequency domain, without relying on modal superposition.

The strategy:

• assembles the mass and stiffness operators
• optionally assembles the damping operator
• builds the complex dynamic matrix
• solves the complex-valued linear system with a complex linear solver
• stores the real and imaginary parts of the displacement separately

Real and imaginary loads

In the direct harmonic formulation, the excitation vector is complex:

$f = f_{r} + i f_i$

To assemble this consistently, the implementation uses two different load SubModelParts:

• a real load submodelpart, containing the conditions that contribute to $f_{r}$
• an imaginary load submodelpart, containing the conditions that contribute to $f_{i}$

This separation is needed because the direct harmonic strategy assembles the real and imaginary RHS contributions independently and then combines them into the final complex right-hand side.

Imaginary point load condition

A new ImaginaryPointLoadCondition was introduced to assemble imaginary load contributions independently from the real ones.

This makes it possible to define:

• real harmonic loads through the standard point load condition
• imaginary harmonic loads through a dedicated condition

Tests

A new test was added for the direct harmonic analysis strategy.

Included in this PR

• DirectHarmonicAnalysisStrategy
• ImaginaryPointLoadCondition
• support for separate real and imaginary load submodelparts
• new direct harmonic analysis test

[philbucher]

cool stuff!

Quirin was also working on this I think, he was struggling with complex dofs, see #6238

I am curious as to how you solved this, will take a look in the next days (dont wait for me if urgent)

[juancamarotti]

cool stuff!

Quirin was also working on this I think, he was struggling with complex dofs, see #6238

I am curious as to how you solved this, will take a look in the next days (dont wait for me if urgent)

Basically, the builder and solver is providing $M$, $K$ and $C$ to the strategy, and the assembly of the complex RHS is done inside the DirectHarmonicAnalysisStrategy through 2 different SubModelParts, one for the real component of the load and the other for the complex component. Then, we build the assemble the complex LHS/RHS and use a ComplexSolver.
It´s quite straightforward!