Parameter calibration

Table of Contents

• Problem definition
• Calibration

Problem definition

Suppose we are given the experimental data of one pair of strain ${\mathit{\epsilon }}^{\ast }$ and stress ${\mathit{\sigma }}^{\ast }$ measurement of a specimen, and we are tasked to infer the bulk $K$ and shear $G$ modulus of the material.

Let us define the loss function as the distance between the stress prediction and the stress measurement, i.e.

$$l(K,G)={\Vert \mathit{\sigma }({\mathit{\epsilon }}^{\ast };K,G)-{\mathit{\sigma }}^{\ast }\Vert }^2.$$

The material parameters, $K^{\ast }$ and $G^{\ast }$, can then be "inferred" by minimizing the loss function

$$(K^{\ast },G^{\ast })=\underset{K,G}{\mathrm{argmin}}l.$$

Calibration

The following Python script uses the simple gradient descent algorithm to find $K^{\ast }$ and $G^{\ast }$:

$$\begin{array}{r}{K}_{i+1}=K_i-\gamma \frac{\partial l}{\partial K_i},G_{i+1}=G_i-\gamma \frac{\partial l}{\partial G_i},\end{array}$$

which iterates until the loss function $l$ is sufficiently small. A constant learning rate $\gamma =1$ is chosen in this example.

import neml2
from neml2.tensors import SR2
import torch
from matplotlib import pyplot as plt
 
torch.set_default_dtype(torch.double)
model = neml2.load_model("input.i", "model")
 
# Experimental data
strain_exp = SR2.fill(0.1, 0.05, -0.03, 0.02, 0.06, 0.03)
stress_exp = SR2.fill(2.616, 1.836, 0.588, 0.312, 0.9361, 0.468)
 
# The gradient descent loop
gamma = 1
loss_history = []
K_history = []
G_history = []
for i in range(100):
    # Enable AD
    model.K.requires_grad_()
    model.G.requires_grad_()
 
    # Calculate loss
    stress = model.value({"forces/E": strain_exp})["state/S"]
    l = torch.linalg.norm(stress.torch() - stress_exp.torch()) ** 2
 
    # Record state
    loss_history.append(l.item())
    K_history.append(model.K.torch().item())
    G_history.append(model.G.torch().item())
 
    # Update parameters
    l.backward()
    model.K = model.K.tensor().detach() - gamma * model.K.grad
    model.G = model.G.tensor().detach() - gamma * model.G.grad
 
# Make plots
fig, axl = plt.subplots(figsize=(8, 5))
axl.plot(torch.arange(100), loss_history, "k-")
axl.set_xscale("log")
axl.set_yscale("log")
axl.set_xlabel("Iteration")
axl.set_ylabel("Loss")
 
axp = axl.twinx()
axp.plot(torch.arange(100), K_history, "r--", label="Bulk modulus")
axp.plot(torch.arange(100), G_history, "b--", label="Shear modulus")
axp.set_ylabel("Parameter value")
axp.legend()
 
fig.tight_layout()
fig.savefig("calibration.svg")

Material parameter calibration

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