Problems¶
Topology optimization problem to solve.
Base Problem¶
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class
topopt.problems.Problem(bc, penalty)[source]¶ Abstract topology optimization problem.
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bc¶ The boundary conditions for the problem.
Type: BoundaryConditions
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f¶ The right-hand side of the FEM equation (forces).
Type: numpy.ndarray
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u¶ The variables of the FEM equation.
Type: numpy.ndarray
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obje¶ The per element objective values.
Type: numpy.ndarray
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__init__(bc, penalty)[source]¶ Create the topology optimization problem.
Parameters: - bc (
BoundaryConditions) – The boundary conditions of the problem. - penalty (
float) – The penalty value used to penalize fractional densities in SIMP.
- bc (
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compute_objective(xPhys, dobj)[source]¶ Compute objective and its gradient.
Parameters: Returns: The objective value.
Return type:
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Compliance Problem¶
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class
topopt.problems.ComplianceProblem(bc, penalty)[source]¶ Topology optimization problem to minimize compliance.
\(\begin{aligned} \min_{\boldsymbol{\rho}} \quad & \mathbf{f}^T\mathbf{u}\\ \textrm{subject to}: \quad & \mathbf{K}\mathbf{u} = \mathbf{f}\\ & \sum_{e=1}^N v_e\rho_e \leq V_\text{frac}, \quad 0 < \rho_\min \leq \rho_e \leq 1\\ \end{aligned}\)
where \(\mathbf{f}\) are the forces, \(\mathbf{u}\) are the displacements, \(\mathbf{K}\) is the striffness matrix, and \(V\) is the volume.
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compute_objective(xPhys, dobj)[source]¶ Compute compliance and its gradient.
The objective is \(\mathbf{f}^{T} \mathbf{u}\). The gradient of the objective is
\(\begin{align} \mathbf{f}^T\mathbf{u} &= \mathbf{f}^T\mathbf{u} - \boldsymbol{\lambda}^T(\mathbf{K}\mathbf{u} - \mathbf{f})\\ \frac{\partial}{\partial \rho_e}(\mathbf{f}^T\mathbf{u}) &= (\mathbf{K}\boldsymbol{\lambda} - \mathbf{f})^T \frac{\partial \mathbf u}{\partial \rho_e} + \boldsymbol{\lambda}^T\frac{\partial \mathbf K}{\partial \rho_e} \mathbf{u} = \mathbf{u}^T\frac{\partial \mathbf K}{\partial \rho_e}\mathbf{u} \end{align}\)
where \(\boldsymbol{\lambda} = \mathbf{u}\).
Parameters: Returns: The compliance value.
Return type:
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Time-Harmonic Loads Problem¶
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class
topopt.problems.HarmonicLoadsProblem(bc, penalty)[source]¶ Topology optimization problem to minimize dynamic compliance.
Replaces standard forces with undamped forced vibrations.
\(\begin{aligned} \min_{\boldsymbol{\rho}} \quad & \mathbf{f}^T\mathbf{u}\\ \textrm{subject to}: \quad & \mathbf{S}\mathbf{u} = \mathbf{f}\\ & \sum_{e=1}^N v_e\rho_e \leq V_\text{frac}, \quad 0 < \rho_\min \leq \rho_e \leq 1\\ \end{aligned}\)
where \(\mathbf{f}\) is the amplitude of the load, \(\mathbf{u}\) is the amplitude of vibration, and \(\mathbf{S}\) is the system matrix (or “dynamic striffness” matrix) defined as
\(\begin{aligned} \mathbf{S} = \mathbf{K} - \omega^2\mathbf{M} \end{aligned}\)
where \(\omega\) is the angular frequency of the load, and \(\mathbf{M}\) is the global mass matrix.
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__init__(bc, penalty)[source]¶ Create the topology optimization problem.
Parameters: - bc (
BoundaryConditions) – The boundary conditions of the problem. - penalty (
float) – The penalty value used to penalize fractional densities in SIMP.
- bc (
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build_M(xPhys, remove_constrained=True)[source]¶ Build the stiffness matrix for the problem.
Parameters: Returns: The stiffness matrix for the mesh.
Return type:
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build_indices()[source]¶ Build the index vectors for the finite element coo matrix format.
Return type: None
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compute_displacements(xPhys)[source]¶ Compute the amplitude of vibration given the densities.
Compute the amplitude of vibration, \(\mathbf{u}\), using linear elastic finite element analysis (solving \(\mathbf{S}\mathbf{u} = \mathbf{f}\) where \(\mathbf{S} = \mathbf{K} - \omega^2\mathbf{M}\) is the system matrix and \(\mathbf{f}\) is the force vector).
Parameters: xPhys ( ndarray) – The element densisities used to build the stiffness matrix.Returns: The displacements solve using linear elastic finite element analysis. Return type: numpy.ndarray
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compute_objective(xPhys, dobj)[source]¶ Compute compliance and its gradient.
The objective is \(\mathbf{f}^{T} \mathbf{u}\). The gradient of the objective is
\(\begin{align} \mathbf{f}^T\mathbf{u} &= \mathbf{f}^T\mathbf{u} - \boldsymbol{\lambda}^T(\mathbf{K}\mathbf{u} - \mathbf{f})\\ \frac{\partial}{\partial \rho_e}(\mathbf{f}^T\mathbf{u}) &= (\mathbf{K}\boldsymbol{\lambda} - \mathbf{f})^T \frac{\partial \mathbf u}{\partial \rho_e} + \boldsymbol{\lambda}^T\frac{\partial \mathbf K}{\partial \rho_e} \mathbf{u} = \mathbf{u}^T\frac{\partial \mathbf K}{\partial \rho_e}\mathbf{u} \end{align}\)
where \(\boldsymbol{\lambda} = \mathbf{u}\).
Parameters: Returns: The compliance value.
Return type:
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static
lm(nel)[source]¶ Build the element mass matrix.
\(M = \frac{1}{9 \times 4n}\begin{bmatrix} 4 & 0 & 2 & 0 & 1 & 0 & 2 & 0 \\ 0 & 4 & 0 & 2 & 0 & 1 & 0 & 2 \\ 2 & 0 & 4 & 0 & 2 & 0 & 1 & 0 \\ 0 & 2 & 0 & 4 & 0 & 2 & 0 & 1 \\ 1 & 0 & 2 & 0 & 4 & 0 & 2 & 0 \\ 0 & 1 & 0 & 2 & 0 & 4 & 0 & 2 \\ 2 & 0 & 1 & 0 & 2 & 0 & 4 & 0 \\ 0 & 2 & 0 & 1 & 0 & 2 & 0 & 4 \end{bmatrix}\)
Where \(n\) is the total number of elements. The total mass is equal to unity.
Parameters: nel ( int) – The total number of elements.Returns: The element mass matrix for the material. Return type: numpy.ndarray
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von Mises Stress Problem¶
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class
topopt.problems.VonMisesStressProblem(nelx, nely, penalty, bc, side=1)[source]¶ Topology optimization problem to minimize stress.
Todo
- Currently this problem minimizes compliance and computes stress on the side. This needs to be replaced to match the promise of minimizing stress.
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static
B(side)[source]¶ Construct a strain-displacement matrix for a 2D regular grid.
\(B = \frac{1}{2s}\begin{bmatrix} 1 & 0 & -1 & 0 & -1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 & 0 & -1 & 0 & -1 \\ 1 & 1 & 1 & -1 & -1 & -1 & -1 & 1 \end{bmatrix}\)
where \(s\) is the side length of the square elements.
Todo
- Check that this is not -B
Parameters: side ( float) – The side length of the square elements.Returns: The strain-displacement matrix for a 2D regular grid. Return type: numpy.ndarray
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static
E(nu)[source]¶ Construct a constitutive matrix for a 2D regular grid.
\(E = \frac{1}{1 - \nu^2}\begin{bmatrix} 1 & \nu & 0 \\ \nu & 1 & 0 \\ 0 & 0 & \frac{1 - \nu}{2} \end{bmatrix}\)
Parameters: nu – The Poisson’s ratio of the material. Returns: The constitutive matrix for a 2D regular grid. Return type: numpy.ndarray
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__init__(nelx, nely, penalty, bc, side=1)[source]¶ Create the topology optimization problem.
Parameters: - bc – The boundary conditions of the problem.
- penalty – The penalty value used to penalize fractional densities in SIMP.
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static
dsigma_pow(s11, s22, s12, ds11, ds22, ds12, p)[source]¶ Compute the gradient of the stress to the \(p^{\text{th}}\) power.
\(\nabla\sigma^p = \frac{p\sigma^{p-1}}{2\sigma}\nabla(\sigma^2)\)
Todo
- Properly document what the sigma variables represent.
- Rename the sigma variables to something more readable.
Parameters: Returns: The gradient of the von Mises stress to the \(p^{\text{th}}\) power.
Return type:
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static
sigma_pow(s11, s22, s12, p)[source]¶ Compute the von Mises stress raised to the \(p^{\text{th}}\) power.
\(\sigma^p = \left(\sqrt{\sigma_{11}^2 - \sigma_{11}\sigma_{22} + \sigma_{22}^2 + 3\sigma_{12}^2}\right)^p\)
Todo
- Properly document what the sigma variables represent.
- Rename the sigma variables to something more readable.
Parameters: Returns: The von Mises stress to the \(p^{\text{th}}\) power.
Return type: