Surrogate Modeling

Additive Manufacturing (AM) is intensifying the digitalization of the manufacturing and
generating disruptive changes and the paradigm shift in the industry. To boost the productivity
in AM, the improvement in parts qualification is required. Qualifying the parts implies the
capability to ensure constant and repeatable part properties meeting the engineering design
requirements. It implies to certify not only the structure, properties and global performance of a
part but also its manufacturing process. Modelling and simulation techniques can target them
separately, but linking models describing those characteristics is quite challenging since they
have the different level of details and sometimes different purposes. Presenting those models in
the form of causal graph will enable us to combine models with different level of details. The
current research utilizes functional analysis and dimensional analysis to present the models in
the form of the causal graph between associated parameters. The DACM Framework integrating
fundamental required methods and theories is briefly explained in this article. The current
research will then aim at developing a surrogate model capable of integrating the different
models such as microstructure models, layer by layer melting/solidification and part models. The
final comprehensive model can simulate to AM system to fulfill multi-criteria performances

In engineering design, surrogate models are often used instead of costly computer simulations. Typically, a single surrogate model is selected based on the previous experience. We observe, based on an analysis of the published literature, that fitting an ensemble of surrogates (EoS) based on cross-validation errors is more accurate but requires more computational time. In this paper, we propose a method to build an EoS that is both accurate and less computation-ally expensive. In the proposed method, the EoS is a weighted average surrogate of response surface models, kriging, and radial basis functions based on overall cross-validation error. We demonstrate that created EoS is accurate than individual surrogates even when fewer data points are used, so computationally efficient with relatively insensitive predictions. We demonstrate the use of an EoS using hot rod rolling as an example. Finally, we include a rule-based template which can be used for other problems with similar requirements, for example, the computational time, required accuracy, and the size of the data. Frame of reference Computer simulations are commonly used to replace experiments with physical models. Often these simulations are computationally expensive. However, many model-based engineering design problems require numerous simulations to reach an acceptable solution. This can be computationally prohibitive. Often a single surrogate model-or metamodel-is used to replace a detailed simulation in design problems which require repeated calculations. This surrogate is obtained using information derived from the physical model. Many types of surrogate models have been proposed. Here, we demonstrate the advantages of computing ensemble of surrogates (EoS) from a single set of data and then averaging these surrogates to make use of the good features of each type of surrogate. We term these assemblies of surrogates, ensembles of surrogates. The EoS is built using the weighted average of different individual surrogates. These weights can be calculated randomly or in a systematic way. For instance, Viana and Haftka (2008) use a systematic weighted average surrogate (WAS) to utilize the advantage of n surrogates to cancel the errors in estimation by appropriate weight selection in the linear mix of the models. They use an ensemble of metamodels to minimize the root-mean-square error (RMSE) in surrogate mod-eling. They discuss using the lowest predicted residual error sum of squares (PRESS) solution or just an average surrogate when an individual surrogate is required. They also propose the optimization of the integrated squared error (SE) as an approach to calculate the weights of the WAS. They found that it is worthwhile to create a broad set of various surrogates and after that apply PRESS as the selection criterion. Cross-validation is utilized broadly to allocate the weights to individual surrogates in building an EoS in a systematic way. Viana et al. (2009) use cross-validation to approximate the necessary safety border for a particular favorite conservativeness degree (safe approximations percentage). They also check how well they can reduce the loss of accuracy caused by a conservative estimator 1 by choosing among alternative surrogate models. They show that cross-validation enables to choose the best conservative surrogate model with the lowest accuracy loss. Also, they found that it is efficient in determining safety is effective for selecting the safety edge. Also, Viana et al. compare using the lowest PRESS with a weighted surrogate when an individual surrogate is required. 1 The estimator is the PRESS and obtained by dividing the N data points into k subsets in cross-validation process. It is estimated by using a subset of points in building the surrogate and computing the errors at these left out points. Then, this process is replicated with various sets of left-out points to obtain PRESS which is statistically significant.

Sobol' indices are a common metric of dependency in sensitivity analysis. It is used as a measure of confidence of input variables influence on the output of the analyzed mathematical model. We consider a problem of selection of experimental design points for Sobol' indices estimation. Based on the concept of D-optimality, we propose a method for constructing an adaptive design of experiments, effective for the calculation of Sobol' indices from Polynomial Chaos Expansions. We provide a set of applications that demonstrate the efficiency of the proposed approach.

Problem of aircraft structural components (wing, fuselage, tail) optimization is considered. Solution of this problem is very computationally intensive, since it requires at each iteration a two-level process. First from previous iteration an update step at full component level must be performed in order to take into account internal loads and their sensitivities in the whole structure involved by changes in local geometry. Second numerous local analyzes are run on isolated elements (for example, super stiffeners) of structural components in order to calculate mechanical strength criteria and their sensitivities depending on current internal loads. An optimization step is then performed from combined global-local sensitivities. This bi-level global-local optimization process is then repeated until convergence of load distribution in the whole structure. Numerous calculations of mechanical strength criteria are necessary for local analyzes and results in great increase of the time between two iterations. In this work an effective method for speeding up the optimization process was elaborated. The method uses surrogate models of optimization constraints (mechanical strength criteria) and provides reduction of the structure optimization computational time from several days to a few hours.

Metamodels are a valuable tool to support a wide scope of activities in modern engineering design, especially design optimization, and allows to get products with prescribed level of reliability at a significantly lower time compared to expensive simulation processes. In this work we construct metamodels for approximation of optimization constraints by performing regression of static mechanical criteria, namely buckling and collapse reserve factors of a stiffened panel. Due to the highly non-linear behavior of these functions with respect to loadings and design variables the use of conventional methods of approximation (Kriging, Artificial Neural Networks, Support Vector Regression, Multivariate Nonparametric Regression, Polynomial Regression) did not allow us to get sucient quality on the whole design space. We show how metamodel with very high quality can be constructed on basis of newly developed approximation technique NARB (Neural Approximation based on Regression and Boosting).

We consider a problem of adaptive design of experiments for Gaussian process regression. We introduce a Bayesian framework, which provides theoretical justification for some well-know heuristic criteria from the literature and also gives an opportunity to derive some new criteria. We also perform testing of methods in question on a big set of multidimensional functions.

We describe GTApprox — a new tool for medium-scale surrogate modeling in industrial design. Compared to existing software, GTApprox brings several innovations: a few novel approximation algorithms, several advanced methods of automated model selection, novel options in the form of hints. We demonstrate the efficiency of GTApprox on a large collection of test problems. In addition, we describe several applications of GTApprox to real engineering problems.