156 S. Myren et al. Table 16.1 Parameters and ranges considered in the compressive tantalum model Parameter Units Description Min. Max. C1 – Taylor hardening coefficient 0.3 1 B0 MBar-μs Reference phonon drag viscosity 2e−11 1e−9 g0 – Dislocation barrier energy coefficient 0.01 1 ω0 MHz Dislocation attempt frequency 0.04375 8.75 p – Energy barrier shape coefficient 0.1 2 q – Energy barrier shape coefficient 0.1 2 τ0 stress-MBar Intrinsic lattice resistance 1e−06 1e−2 Varrhom0 ( M0) cm−2 Initial mobile dislocation density 1e8 1e12 Varrhoi0( i0) cm−2 Initial immobile dislocation density 1e1 1e12 L cm Mean spacing between barriers 1e−5 1e−4 CM – Dislocation multiplication coefficient 0.01 10 CA – Dislocation annihilation coefficient 0.1 1000 CT – Dislocation trapping coefficient 0.1*CM 1*CM Rhosati SAT i cm−2 Saturation density of immobile dislocations Varrhoi0 Varrhoi0 +1e15 16.4 Methodology 16.4.1 Bayesian Statistics and Estimation We adopt a Bayesian approach to the problem of estimating the best fitting input parameters for the computational materials model. A Bayesian statistical model has two parts, a likelihood and a prior distribution. The likelihood describes the probability distribution of the data given the unknown materials parameters. The prior distribution describes our best guess at a probability distribution for the unknown parameters before considering the current data. To build our likelihood, we assume that the experimental data are a noisy version of the materials model run at the best fitting parameters. The materials model at a given parameter setting returns a stress-strain curve. Our experiments also produce such a curve. We will assume that the experimental curve should be a materials model curve with Gaussian error at each point. Denote our experimental curve by y, our unknown best fitting materials parameters byθ, the materials model by η(·), and our Gaussian error variance as σ 2. We can write y ∼N5η(θ),σ 2 I6 (16.4) which says that the vector y is multivariate normal with mean vector given by η(θ) and independent noise at each location with variance σ 2. Our prior distribution for the unknown parameter vector θ is just uniform over a fixed range for each parameter independently. The ranges are given in Table 16.1 and are chosen based on expert knowledge about the materials properties and model, derived in [3, 4]. The posterior distribution is the probability distribution of the unknown parameters given the observed experimental data. It is proportional to the product of the prior and the likelihood p(θ|y) =f (y|θ)I {θ ∈H}/K (16.5) where f(y|θ) is the Gaussian likelihood described above and I{θ ∈ H} is the uniform prior distribution. The value K is a normalizing constant that typically cannot be computed because it involves the solution to a difficult integral. This means it is hard to compute things like the mean and variance directly. It is also difficult to sample from this distribution using straightforward approaches. Instead, we use an approach called Markov chain Monte Carlo (MCMC). MCMC is a sequential sampling procedure that produces a correlated sample from a specified distribution. The algorithm is described in Fig. 16.3. Because the unknown normalizing constant appears in both the numerator and denominator of the acceptance probability of Step 2, it cancels out and we do not ever need to compute it. This approach only requires forward evaluations of the posterior; we do not to invert it or compute derivatives. Thus, we can use this approach with the compressive tantalum materials model simply by computing the output at any candidate input from Step 1. One disadvantage of this algorithm is that it may requires tens of thousands of correlated samples in order to produce good estimates. This can be intractably slow when used with a materials model, even if the simulation requires only a few
RkJQdWJsaXNoZXIy MTMzNzEzMQ==