Bayesian optimization (BO) for origami-inspired folding structures

• 1D schematic
  

1. Upper panel
• Goal : To optimize expensive function (black dashed curve) in the upper panel
• Surrogate : Gaussian process (GP) surrogate model is trained with the points (black dots) and is stochastic, with the uncertainty levels (blue bands) and the mean of GP (solid blue curve) in the upper panel.
2. Lower panel
• Scalarization : Acquisition function (green solid curve) in the lower panel scalarizes the surrogate model to give single-valued function
• Optimization : Acquisition function (green solid curve) is minimized to find the next training point shown with the red triangle.
• Iteration : The expensive function is evaluated at the new training point (red circle) and added to the training data. The process is repeated with the surrogate model mimicking the expensive function better with each iteration. The uncertainty in the surrogate model is also reduced as more training points are added.

• Chomper origami folding structure
  

Problem setup	Evolution of objective function
Number of objective function evaluations	Best fold pattern

• Traditional gradient-based optimization technique gets trapped in local minima.
• Bayesian (BO) finds optimal design and takes ~34 x fewer FE solution than genetic algorithm (GA)

• Twist chomper origami folding structure
  

Problem setup	Evolution of objective function
Number of objective function evaluations	Best fold pattern

• Traditional gradient-based optimization technique gets trapped in local minima.
• Bayesian (BO) finds optimal design and takes ~56 x fewer FE solution than genetic algorithm (GA)

Companion paper: Bayesian topology optimization for efficient design of origami folding structures

Gradient-enhanced Bayesian optimization (GEBO)

• 1D schematic
  

BO	Gradient enriched BO

• 2D schematic
  

Objective function	GP	Gradient enriched GP

• Gradient enriched GP matches the objective functions gradients and function values at the training points.
• Gradient enriched GP gives a better fit than traditional GP.

• Airfoil shape optimization
  

Problem schematic	CFD mesh
Evolution of objective function	Computational time

• Objective: To find the optimal airfoil shape to minimize drag.
• GEBO with gradient clipping gives ~3 x speedup than traditional BO.

• Square twist origami folding structure
  

Problem setup	Evolution of objective function
Computational time	Best fold pattern

• Objective: To find the optimal fold pattern to maximize actuation.
• GEBO with gradient clipping gives ~3.5 x speedup than traditional BO.

Companion paper: Systematic cost analysis of gradient- and anisotropy-enhanced Bayesian design optimization

Anisotropy-enhanced Bayesian optimization

• 2D schematic
  

Objective function	Isotropic GP	Anisotropy enriched GP

• Anisotropy-enhanced GP gives a better fit than traditional GP for functions with dominant design variables.
• Anisotropy-enhanced GP can be used as a dimension reduction technique using automatic relevance determination (ARD).

• Twist chomper origami folding structure
  

Problem setup	Evolution of objective function
Computational time	Best fold pattern

• Objective: To find the optimal fold pattern to maximize actuation.
• Anisotropy-enhanced BO with (ARD) reduces the dimension from 38 to 16.
• Anisotropy enhanced BO with (ARD) gives ~2 x speedup than traditional BO.

Companion paper: Systematic cost analysis of gradient- and anisotropy-enhanced Bayesian design optimization

Deep energy minimization (DEM) framework for hyperelastic multistable structures

• Schematic DEM-SF
  

• Solver: A Fully connected feed-forward neural network approximates the displacement field.
• Loss: Potential energy of the system is minimized as loss (DEM).
• Gradients: Finite element shape functions are used to calculate the gradients of displacement (DEM-SF).

• Buckling of rectangular beam
  

FEM Compression	FEM Buckling (Needs Eignevalue solution)	DEM-SF

Minimum potential energy evolution comparison.

• FEM gives compression deformation mode without eigenvalue solution.
• FEM requires an eigenvalue solution for perturbation to give buckling deformation mode.
• The deep energy minimization framework when FE shape function (DEM-SF) is utilized to calculate the gradients gives buckling deformation mode.
• The deep energy minimization framework when automatic differentiation (DEM-AD) is utilized to calculate the gradients gives a non-physical solution.

• Bi-stable unit cell
  

FEM Compression	FEM Buckling (Needs Eignevalue solution)	DEM-SF

Minimum potential energy evolution comparison.

• FEM needs an eigenvalue solution to predict the low potential buckling path.
• DEM-SF finds the low potential buckling path without an eigenvalue solution.
• DEM-SF accurately predicts the second stable state.

• Multistable structure
  

DEM-SF deformation	Minimum potential energy evolution

• DEM-SF can accurately predict multiple stable states.

• Load step independence
  

Problem setup	Minimum potential energy evolution

• DEM-SF finds accurate solutions with larger load steps.
• FEM with larger load steps runs into convergence issues.

• Transfer learning
  

Trained design	New design
Loss history	Minimum potential energy evolution

• Transfer learning can be leveraged for new designs to speed up the solution time by at least 5x.

Deep energy minimization framework for phase-field elastic-perfectly plasticity problems

• Schematic
  

• Solver: A Fully connected feed-forward neural network approximates the displacement and scalar phase field.
• Slip: The scalar phase field, s $\left(0\leq s\leq 1\right)$, called slip represents the degree of plastic slip in the body such that the bulk material response is degraded in the areas where $s>0$ plastic slip.
• Loss: Potential energy of the system and the slip irreversibility condition is minimized as loss.
• Gradients: In-built automatic differentiation library of PyTorch is used to calculate the gradients of displacement and phase field (DEM-AD).

• Extension of square specimen

• Elasticity parameters: Youngs modulus, $E=2.1e5$ ; Poissons ratio, $\nu = 0.3$.
• Plasticity parameters: Unpinning stiffness per area $\gamma = 0.1$; gliding resistance, $\sigma_0 = 120$; characteristic length $\epsilon = 2.5e-2$ .

Problem setup	Phase field s
Mininum potential energy	Reaction force

• FEM with first order fully integrated element gives unrealistic stiff response due to volumetric locking that results in wider shear bands.
• FEM with second-order selective reduce integration (SRI) is needed to counter the volumetric locking issue and it predicts realistic sharper shear bands.
• DEM-AD can provide a volumetric locking-free solution and predicts sharper shear bands.

• Extension of square plate with hole

• Elasticity parameters: Youngs modulus, $E=2.1e5$ ; Poissons ratio, $\nu = 0.3$.
• Plasticity parameters: Unpinning stiffness per area $\gamma = 0.025$; gliding resistance, $\sigma_0 = 120$; characteristic length $\epsilon = 2.5e-2$ .

Problem setup	Phase field s
Mininum potential energy	Reaction force

• FEM and DEM-AD solution accurately predicts the onset of the plastic slip near the hole.
• FEM with second-order selective reduce integration (SRI) follows the higher minimum potential energy path indicating that it gets trapped in local solution.
• DEM-AD follows the lower minimum potential energy path.

• Extension of rectangular plate with two asymmetrical notches

• Elasticity parameters: Youngs modulus, $E=2.1e5$ ; Poissons ratio, $\nu = 0.3$.
• Plasticity parameters: Unpinning stiffness per area $\gamma = 0.2$; gliding resistance, $\sigma_0 = 80$; characteristic length $\epsilon = 2.5e-2$ .

Problem setup	Phase field s
Mininum potential energy	Reaction force

• FEM and DEM-AD solution accurately predicts the onset of the plastic slip near the notches.
• FEM with second-order selective reduce integration (SRI) follows the higher minimum potential energy path that leads to wider shear bands.
• DEM-AD follows the lower minimum potential energy path that leads to sharper shear bands.

• Unique calibration of plasticity parameters

BO loss	$\gamma$ (True value $= 0.1$)	$\sigma_0$ (True value $= 120$)	$\epsilon$ (True value $= 2.5e-2$)

BO loss	$\gamma$ (True value $= 0.3$)	$\sigma_0$ (True value $= 70$)	$\epsilon$ (True value $= 4e-2$)

• Bayesian optimization (BO) with displacement data obtained using digital image correlation (DIC) is considered for the calibration of the plasticity parameters.
• Anisotropy is embedded in the Bayesian optimization (BO) framework by using unique kernel length scales for each plasticity parameter.
• The neural network is initially trained on the DIC displacement data to get a good starting point for the weights
• DEM-AD equipped with transfer learning is utilized to solve the necessary forward problems.
• DEM-AD equipped with transfer learning uniquely calibrates plasticity parameters 6x faster than using FEM.