[Structural] Add system for generic pre-tensioning

[matekelemen]

Copying the docs here because they explain the same thing.

Pre-Tensioning

Apply pre-tensioning to 1D, 2D, or 3D structural parts.

Pre-tensioning approximately models connector parts (e.g., bolts, screws, etc.) subject to initial loading within an assembly. It is intended as a simplified alternative to high-fidelity analyses of such assemblies, placing less emphasis on the connector itself while still capturing similar effects on the surrounding structure.

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Pre-Tensioning Surface

The pre-tensioning system acts on a surface defined by a set of geometries within a model part. The following requirements apply:

• They must be defined on element boundaries (faces of polyhedra, edges of polygons, vertices of lines).
• They must partition the connector part's elements into two distinct sets.
• They should lie on the same plane as much as possible.
• They must be stored in a separate sub model part containing only these geometries.

A pre-tensioning plane is computed by averaging the pre-tensioning surface. The normal of this plane serves as the basis for displacement constraints introduced later.

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Node Duplication

The pre-tensioning surface partitions adjacent elements into two groups:

• Positive side
• Negative side

Nodes lying on the surface are duplicated. Nodes belonging to negative-side elements are replaced with duplicates, while positive-side elements remain unchanged.

This effectively cuts the connector part in half along the pre-tensioning surface. Duplicated nodes are inserted into the sub model part containing the surface geometries.

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In-Plane Constraints

After duplicating nodes, constraints are introduced to restrict displacements and rotations.

If nodes have rotational degrees of freedom, the rotations of duplicates must match their originals:

$$\phi_o^i - \phi_d^i = 0 \quad \forall \quad 1 \leq i \leq m$$

Where:

• $\phi_o^i$: rotation of the original node $i$
• $\phi_d^i$: rotation of the duplicate of node $i$
• $m$: number of original nodes on the surface

Additionally, the in-plane displacement difference must vanish:

$$u_o^i - u_d^i - \left< u_o^i - u_d^i, n \right> n = 0 \quad \forall \quad 1 \leq i \leq m$$

Where:

• $u_o^i$: displacement of the original node $i$
• $u_d^i$: displacement of the duplicate node $i$
• $n$: unit normal of the pre-tensioning plane
• $\langle \cdot , \cdot \rangle)$: inner product

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Out-of-Plane Constraints

To apply a single pre-tensioning value (force or displacement), the out-of-plane displacement components are averaged and tied to virtual degrees of freedom:

$$\begin{align} u_v^o - \frac{1}{m} \sum_{i=1}^m{ \left< u_o^i, n \right> } &= 0 \\\ u_v^d - \frac{1}{m} \sum_{i=1}^m{ \left< u_d^i, n \right> } &= 0 \end{align}$$

Where:

• $u_v^o$: virtual DoF for original nodes
• $u_v^d$: virtual DoF for duplicate nodes

These virtual degrees of freedom can then be either loaded (Neumann-type) or fixed (Dirichlet-type).

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Neumann-Type Pre-Tensioning

Given a pre-tensioning force $f$, enforce:

$$\begin{align} r_v^o - f &= 0 \\ r_v^d + f &= 0 \end{align}$$

Where:

• $r_v^o$: reaction of $u_v^o$
• $r_v^d$: reaction of $u_v^d$

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Dirichlet-Type Pre-Tensioning

Fix the relative average out-of-plane displacement to a prescribed value $\alpha$:

$$u_v^o - u_v^d - \alpha = 0$$

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Implementation Notes

In-plane displacement constraints reuse degrees of freedom and are dense. They cannot be imposed via master-slave elimination unless DoFs are rotated to align with the pre-tensioning plane.

Since there is no robust way to perform this rotation in Kratos, constraints must be enforced using the augmented Lagrange multiplier method, implemented in PMultigridBuilderAndSolver.

Note: Any analysis involving pre-tensioning must use PMultigridBuilderAndSolver.

[philbucher]

Interesting, this appears to be similar to bolt preloads

Are there plans to port the ALM also to the standard B&S?

[matekelemen]

Yes I think they should be rather similar. Abaqus also has a system for doing this kind of analysis.

Long story short, I don't think ALM will be supported with other B&S types unfortunately.

The TC is adamant about keeping constraints strictly master-slave imposed, meaning that other types of impositions (penalty, lagrange, ALM) have to be implemented as elements. This has several unsavory consequences:

1. every constraint type has to be implemented for each kind of imposition method separately,
2. the ALM loop, which normally would only require updating a subset of the RHS coming from constraints, can only exist as part of a full-blown non-linear analysis. This not only means that a complete assembly must happen in every iteration, but also prohibits taking advantage of linear solvers that benefit from pre-factorizing the LHS matrix,
3. changing imposition methods requires major changes to the analysis (different meshes with different constraint elements, change between linear and non-linear analyses, etc.)

These were all deal breakers for me, so I wrote my own B&S instead. The silver lining is that PMGB&S can do everything the standard B&S can, except MPI (for now).