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Worst-Case Stress Optimization

Executable: worst_case_stress/WCSOptimization_cli

This package implements the worst-case microstructure stress analysis and minimization, besides having also the same tools presented in the initial pattern optimization technique.

Main parameters

  • Choose target properties of material, like desired elasticity tensor $$C^*$$, Young's module, Poisson's ration and if it is isotropic. The target file is the single input of the method.

    Also, this option let us set initial values for the parameters of the structure (like offset and vertices radius) and apply bound constraints on them.

    Here is an example of a job.opt serving as target:

    {
  "dim": 2,
  "target": {
      "type": "isotropic",
      "young":   10,
      "poisson": 0.35
  },
  "initial_params": [0.6809516712393716, 0.6280027237433981, 0.5640910410666709, 0.4337430787476562,
      0.08129922564391019, 0.1842173643713042, 0.1325967598554578, 0.1750543806991497,
      0.1945952629563027, 0.009638340078114519, 0.01, 0.01, 0.00826325588295466, 0.009238195522149741],
  "radiusBounds": [0.04, 0.2],
  "translationBounds": [0.1, 0.8],
  "blendingBounds": [0.0078125, 0.2]
}

    Notice that setting as target only the young and poisson values is enough because we are dealing with an isotropic target. However, we can also set a general elasticity tensor and make the type as "anisotropic". See the two examples below, where one is for 2D cases and the second for 3D cases.

    Examples:

    • 2D:
    {
    "dim": 2,
    "target": { "type": "anisotropic",
    "material_matrix": [[1, 2, 3],
                        [2, 1, 2],
                        [3, 2, 0.5]]
    },
    "initial_params": [0.6809516712393716, 0.6280027237433981, 0.5640910410666709,
        0.4337430787476562, 0.08129922564391019, 0.1842173643713042, 0.1325967598554578, 0.1750543806991497,
        0.1945952629563027, 0.009638340078114519, 0.01, 0.01, 0.00826325588295466, 0.009238195522149741],
    "radiusBounds": [0.04, 0.2],
    "translationBounds": [0.1, 0.8],
    "blendingBounds": [0.0078125, 0.2]
}
    • 3D:
    {
    "dim": 3,
    "target": { "type": "anisotropic",
    "material_matrix": [[1, 0, 0, 0, 0, 0],
                        [0, 1, 0, 0, 0, 0],
                        [0, 0, 1, 0, 0, 0],
                        [0, 0, 0, 0.5, 0, 0],
                        [0, 0, 0, 0, 0.5, 0],
                        [0, 0, 0, 0, 0, 0.5]]
    },
    "initial_params": [0.5, 0.3333333333333332593, 0.6666666666666665186, 0.5, 0.5, 0.5, 0.5, 0.5,
        0.3333333333333332593, 0.6666666666666665186, 0.5, 0.5, 0.5, 0.5, 0.5, 0.6666666666666665186,
        0.3333333333333332593, 0.5, 0.5, 0.6666666666666665186, 0.3333333333333332593,
        0.3333333333333332593, 0.6666666666666665186, 0.5, 0.5, 0.6666666666666665186,
        0.3333333333333332593, 0.07000000000000000666, 0.07000000000000000666, 0.07000000000000000666,
        0.07000000000000000666, 0.07000000000000000666, 0.07000000000000000666, 0.07000000000000000666,
        0.07000000000000000666, 0.07000000000000000666, 0.07000000000000000666, 0.07000000000000000666,
        0.07000000000000000666, 0.07000000000000000666, 0.07000000000000000666, 0.07000000000000000666,
        0.01000000000000000021, 0.01000000000000000021, 0.01000000000000000021, 0.01000000000000000021,
        0.01000000000000000021, 0.01000000000000000021, 0.01000000000000000021, 0.01000000000000000021,
        0.01000000000000000021, 0.01000000000000000021, 0.01000000000000000021, 0.01000000000000000021,
        0.01000000000000000021, 0.01000000000000000021, 0.01000000000000000021],
    "radiusBounds": [0.04000000000000000083, 0.2000000000000000111],
    "translationBounds": [0.1000000000000000056, 0.8000000000000000444],
    "blendingBounds": [0.005000000000000000104, 0.2000000000000000111]
}

  • Choose the pattern to be used with -p <parameter>. The parameter should be a file containing information about the topology of the structure, like a graph, containing vertices and how they are linked by edges. Examples can be found in microstructures/patterns/3D/reference_wires.

    In order to visualize the pattern, you can convert the file (.wire) extension into a .msh file, which makes it possible to use gmsh on it. To make the conversion, use the executable in MeshFEM repository called mesh_convert as shown below:

    ./mesh_convert input.wire output.msh

  • Choose base material with -m <parameter>. Here, most of the times we will use the file B9Creator.material, which contains information about the base material used in the 3D printer, like density, Young's module and Poisson's ratio. Also, it says if the material is isotropic or not. This file can be found at microstructures/materials folder.

  • Choose settings of meshing used in the process through -M <parameter>. With this option, we can define some parameters like limits on edge sizes, ratio between cell radius and edge size and max area of a face.

  • Decide between using vertex thickness (-V) or edge thickness. The radius of each vertex affect directly the edge thickness, since the convex hull of the vertices (which are like balls) is used to define the edges.

  • Set the weights for elements considered in the objective function. The most common parameters are:

    • --WCSWeight <parameter> decides the weight given for minimizing the worst case stress in the mesh. ($$J_{wcs} = (\int_\Omega s^p)^{\frac{1}{p}}$$, where $$p$$ can also be set with --usePthRoot and --pnorm <parameter> options. See below).

    • --JSWeight <parameter> decides the weight given to $$\lVert S(w) - S^* \rVert$$, where $$ S^* $$ is the desired compliance tensor. There is also a constraint $$C(w) = C^*$$ in the optimization problem that can be set using option --TensorFitConstraint.

  • Choose the norm used for the worst case stress, by using --pnorm <parameter> plus --usePthRoot options.

  • Select the solver for the opt problem using --solver <parameter>. Usually, use slsqp.

  • Set prefix name for output of each iteration with -o <parameter>. The idea is that you should be able to see the result of each iteration, for example, by running gmsh on the output. For each iteration, a msh file <parameter>_X (where X is the iteration number) is created.

Complete example

./WCSOptimization_cli -p $MICRO_DIR/docs/examples/octa_cell.obj -m $MICRO_DIR/docs/materials/B9Creator.material \
        target_tensor_job.opt  -M 2d_meshing_opts.opt --ortho_cell --vertexThickness \
        --WCSWeight 1e-300 --JSWeight 1.0 --TensorFitConstraint \
        --solver slsqp -o it

Use cases

  • In the example above, we use a very low worst-case stress objective weight and set the compliance tensor-fitting weight at 1.0. This ensures worst-case stress is still reported at each iterate, but effectively ignores the stress levels when optimizing (focusing only on fitting the desired elasticity tensor).

  • A different configuration would be the following:

./WCSOptimization_cli -p $MICRO_DIR/docs/examples/octa_cell.obj -m $MICRO_DIR/docs/materials/B9Creator.material \
        target_tensor_job.opt  -M 2d_meshing_opts.opt \
        --ortho_cell --vertexThickness --WCSWeight 1.0 --JSWeight 0.0 --pnorm 8 --usePthRoot \
        --TensorFitConstraint  --solver slsqp -o itHighWCSWeight

This example minimizes the worst-case stress only (the compliance tensor does not appear in the optimization objective). However, the target compliance tensor will still be achieved due to the equality constrained imposed by --TensorFitConstraint.

Per-iteration output statistics

The optimizer outputs a list of statistics for each iterate it evaluates. The precise collection of statistics it reports depends on what objetive terms/constraints are included in the optimization.

Name Description
p Pattern parameter vector
moduli Elastic moduli (assuming an orthotropic material): Young's moduli, followed by Poisson's ratios, followed by shear moduli
anisotropy Anisotropy/Zener ratio (assuming a cubic material): the ratio of the actual shear modulus to the shear modulus of an isotropic material with the same Young's modulus and Poisson's ratio. This is 1 for isotropic tensors.
printable Whether the 3D pattern is printable (1) or not (0). 2D iterates always print 1
Rel elasticity tensor dist $$\frac{\lVert E(\omega) - E^* \rVert_F}{\lVert E^* \rVert_F}$$
JFull The full combined objective being optimized (a weighted combination of the objective terms listed below)
TensorFit violation If the tensor-fitting equality constraint is applied: residual $$S(\omega) - S^*$$ flattened into a 1D vector
jacobian TensorFit row norms If the tensor-fitting constraint is applied: norm of the gradient of each component of residual $$S(\omega) - S^*$$

The following objective-specific terms may appear:

Name Description
JS Value of the compliance tensor-fitting objective: $$\lVert S(\omega) - S^*\rVert_F^2$$
ProximityRegularization Value of $$\lVert p - p^* \rVert^2$$, where $$p^*$$ are the "target" pattern parameters
Max Ptwise WCS The greatest worst-case stress level appearing anywhere in the mesh (Linf norm)
WCS The worst-case stress objective value (Lp norm of worst-case stress field)

For each objective present (JFull, JS, ProximityRegularization, WCS, etc.), the following statistics are printed:

Name Description
normalized <objective> Objective value normalized by the value for the initial iterate.
grad_p <objective> Gradient of the objective with respect to each pattern parameter
‖grad_p <objective>‖ Norm of the objective's gradient with respect to the pattern parameters