TOuU (Topology Optimization under Uncertainty)

We propose the use of stochastic gradient descent algorithms for Topology optimization under uncertainty (TOuU) in (citation). To illustrate the approach, we consider a 3D design problem with a structure being subject to uncertain loading and resting on uncertain bedding. The non-design domain is occupied by a material with random stiffness and represents an uncertain bedding. This non-design domain is clamped at the bottom face. At the center of the top face of the design domain, a point load with random direction is applied. The performance measure of the objective function is the strain energy, and the only constraint is to ensure that the mass-ratio of the structure is no more than 15% of the maximum design mass.

License: GNU GPLv3.0

block_launch.m

This file sets up the problem, initializes the variables, and calls the Adam optimizer. Please change the path variables as necessary, for example, paraview_save_path, code_path, run_path, etc. Also, use mkdir if needed to create those directories.

Variables defined:

• The param structure contains

  • paraview_save_path : Indicates the path to save paraview output
  • code_path : Indicates the path with Matlab codes
  • run_path : Indicates the path, where FEMDOC run results are saved
  • n_call_samples : Number of random samples per iteration
  • n_workers : Number of workers assigned
  • obj_history : Objective history
  • con_history : Constraint history
  • obj_mean_history : History of mean of objective
  • con_mean_history : History of means of constraints
  • obj_var_history : History of variance of objective
  • con_var_history : History of variances of constraints
  • numcon : Number of constraints
  • numvar : Number of optimization variables
  • obj_func : Objective at the current iteration
  • con_func : Constraints at the current iteration
  • obj_func_grad : Gradients of the objective with respect to the optimization variables
  • con_func_grad : Gradients of the constraints with respect to the optimization variables
  • obj_con_lambda : Penalty to enforce constraints

• step_size : learning rate parameter for stochastic gradient descent algorithm

• optiter : Number of iteration

• s : Optimization variables

• sl : Lower bounds for optimization variables

• su : Upper bounds for optimization variables

Functions used:

adam, adagrad

Adam and AdaGrad optimizers are implemented.
Inputs:

• params.eval_name,
• params,s,optiter,
• step_size,
• sl,
• su : defined as above.

Outputs:

• params: updated param structure
• s: optimization variable
• obj_hist: objective history

block_val

Calls two separate functions :
block_call : to run the forward problem, and to evaluate the gradients
mean_plus_var : to estimate the robust objective E[f] + λ Var(f), constraints, and their corresponding gradients

Inputs:

• s,
• params

Outputs: * params : defined above * z : * grad : gradients * objective : * constraint :

block_call

calls block_write to write FEMDOC input files.
Inputs :

• s,
• params

Outputs :

• params : defined above
• obj_realizations : n_call_samples number of realizations of the objective
• con_realizations : n_call_samples number of realizations of the constraints
• obj_grads : corresponding gradients of the objective
• con_grads : corresponding gradient of the constraints

mean_plus_var

Inputs:

• params : defined above
• obj : objective realizations
• con : constraint realizations
• obj_grad : objective gradients
• con_grad : constraint gradients

Outputs:

• params : updated params
• obj_functional : E[f] + λ Var(f) = R
• con_functional : E[g] + λ Var(g) = C
• d_obj_functional : ∇R
• d_con_functional : ∇C

block_write

Writes the FEMDOC input files.
Inputs:

• s : optimization variables
• params : defined above

design_var_info_read_script

Reads the optimization variables initially.
Inputs: params
Outputs: params

Results:

To view the optimized structure go to the paraview_save_path and open the paraview_output.e-s.xx file in Paraview.

Matlab also saves the params structure with objective and constraint histories in adam_output_block_sg.exo.mat inside the for_simultaneous_runs folder.

NOTE:

GCMMA codes can be obtained by contacting Prof. K. Svanberg (KTH Royal Institute of Technology).
For a Python version of the SGD algorithms see https://github.com/CU-UQ/SGD