Problems

Compliant mechanism synthesis problems using topology optimization.

Base Problem

class topopt.mechanisms.problems.MechanismSynthesisProblem(bc, penalty)[source]

Topology optimization problem to generate compliant mechanisms.

\(\begin{aligned} \max_{\boldsymbol{\rho}} \quad & \{u_{\text{out}}=\mathbf{l}^{T} \mathbf{u}\}\\ \textrm{subject to}: \quad & \mathbf{K}\mathbf{u} = \mathbf{f}_\text{in}\\ & \sum_{e=1}^N v_e\rho_e \leq V_\text{frac}, \quad 0 < \rho_\min \leq \rho_e \leq 1, \quad e=1, \dots, N.\\ \end{aligned}\)

where \(\mathbf{l}\) is a vector with the value 1 at the degree(s) of freedom corresponding to the output point and with zeros at all other places.

spring_stiffnesses

The spring stiffnesses of the actuator and output displacement.

Type:numpy.ndarray
Emin

The minimum stiffness of elements.

Type:float
Emax

The maximum stiffness of elements.

Type:float
__init__(bc, penalty)[source]

Create the topology optimization problem.

Parameters:
  • nelx – Number of elements in the x direction.
  • nely – Number of elements in the x direction.
  • penalty (float) – Penalty value used to penalize fractional densities in SIMP.
  • bc (MechanismSynthesisBoundaryConditions) – Boundary conditions of the problem.
build_K(xPhys, remove_constrained=True)[source]

Build the stiffness matrix for the problem.

Parameters:
  • xPhys (ndarray) – The element densisities used to build the stiffness matrix.
  • remove_constrained (bool) – Should the constrained nodes be removed?
Returns:
Return type:

The stiffness matrix for the mesh.
compute_objective(xPhys, dobj)[source]

Compute the objective and gradient of the mechanism synthesis problem.

The objective is \(u_{\text{out}}=\mathbf{l}^{T} \mathbf{u}\) where \(\mathbf{l}\) is a vector with the value 1 at the degree(s) of freedom corresponding to the output point and with zeros at all other places. The gradient of the objective is

\(\begin{align} u_\text{out} &= \mathbf{l}^T\mathbf{u} = \mathbf{l}^T\mathbf{u} + \boldsymbol{\lambda}^T(\mathbf{K}\mathbf{u} - \mathbf{f})\\ \frac{\partial u_\text{out}}{\partial \rho_e} &= (\mathbf{K}\boldsymbol{\lambda} + \mathbf{l})^T \frac{\partial \mathbf u}{\partial \rho_e} + \boldsymbol{\lambda}^T\frac{\partial \mathbf K}{\partial \rho_e} \mathbf{u} = \boldsymbol{\lambda}^T\frac{\partial \mathbf K}{\partial \rho_e} \mathbf{u} \end{align}\)

where \(\mathbf{K}\boldsymbol{\lambda} = -\mathbf{l}\).

Parameters:
  • xPhys (ndarray) – The density design variables.
  • dobj (ndarray) – The gradient of the objective to compute.
Returns:
Return type:

The objective of the compliant mechanism synthesis problem.
static lk(E=1.0, nu=0.3)[source]

Build the element stiffness matrix.

Parameters:
  • E (float) – Young’s modulus of the material.
  • nu (float) – Poisson’s ratio of the material.
Returns:
Return type:

The element stiffness matrix for the material.