Multilevel surrogate modeling approach for optimization problems with polymorphic uncertain parameters

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Abstract

The solution of optimization problems with polymorphic uncertain data requires combining stochastic and non-stochastic approaches. The concept of uncertain a priori parameters and uncertain design parameters quantified by random variables and intervals is presented in this paper. Multiple runs of the nonlinear finite element model solving the structural mechanics with varying a priori and design parameters are needed to obtain a solution by means of iterative optimization algorithms (e.g. particle swarm optimization). The combination of interval analysis and Monte Carlo simulation is required for each design to be optimized. This can only be realized by substituting the nonlinear finite element model by numerically efficient surrogate models. In this paper, a multilevel strategy for neural network based surrogate modeling is presented. The deterministic finite element simulation, the stochastic analysis as well as the interval analysis are approximated by sequentially trained artificial neural networks. The approach is verified and applied to optimize the concrete cover of a reinforced concrete structure, taking the variability of material parameters and the structural load as well as construction imprecision into account.

Introduction

Optimization approaches can help to design engineering structures. The consideration of uncertain parameters quantified by random variables within the structural design process is achieved by reliability-based design optimization approaches, see e.g. [1] for an overview. In general, an optimization problem with uncertain parameters cannot be solved, because the solution of an optimization problem requires deterministic measures of the objective function to be minimized or maximized. Descriptive statistics can be used to describe the uncertainty of a quantity of interest by several measures, such as the mean value, the variance or quantile values. These uncertainty measures can also be combined, which allows one to consider the robustness, e.g. by minimizing or maximizing the mean value and at the same time minimizing the variability of a quantity of interest, see e.g. [2] and [3].

If uncertain parameters are quantified by intervals, the concept of tolerance design can be applied. In this case, interval bounds can be used in the objective function, e.g. by minimizing the upper bound of a quantity of interest (worst case scenario).

Stochastic and non-stochastic models can be combined to polymorphic uncertainty models within structural optimization approaches, see e.g. [4] and [5]. This paper contains optimization approaches, for random variables and intervals, where uncertainty measures have to be fused, e.g. by means of minimizing the upper bound of the mean value of a quantity of interest.

The optimization problems are formulated based on finite element (FE) simulations and solved numerically by a particle swarm optimization (PSO) algorithm [6], which requires to perform Monte Carlo simulations together with optimization-based interval analyses of deterministic structural simulations for a number of optimization runs. This results in a high number (up to billions) of evaluations of the deterministic finite element simulation model.

In order to reduce the computational effort, fast to evaluate surrogate models are required, such as artificial neural networks (ANN). ANN surrogate models have already been developed to replace deterministic simulations within stochastic analyses, see e.g. [7], [8], [9], [10], which is also the most time consuming part of the nested interval and stochastic analyses in the presented application. However, it also makes sense to replace the interval analysis with ANNs, see e.g. [11] or the stochastic analysis, see e.g. [12]. Based on the approach in [12], a multilevel surrogate modeling strategy is introduced in this paper, where three ANNs are trained sequentially to approximate the deterministic simulation, the stochastic analysis and the interval analysis.

After a verification test with a benchmark function, the presented multilevel surrogate modeling strategy is applied to minimize the crack width of a reinforced concrete bridge structure. A nonlinear finite element model is utilized to compute the load bearing capacity and the crack patterns of the structure. Based on the finite element simulation results, three ANN surrogate models are trained to approximate the objective function. Within the optimization, some concrete material parameters and the structural loading are modeled as stochastic a priori parameters and the position of the reinforcement layers are defined as interval design parameters.

Section snippets

Optimization problem

In this paper, optimization problems with stochastic and non-stochastic uncertain parameters are formulated and solved. Within the concept of polymorphic uncertain modeling, it is focused on the combination of random variables and intervals in the input space, which leads to probability boxes of a quantity of interest Z to be maximized or minimized. In general, such an optimization problem can be formulated asmin or max:Z(A,a,d,D,d)(a)subject to:gj(A,a,d,D,d)0,j=1,,J(b)hk(A,a,d,D,d)=0,

Multilevel surrogate modeling strategy

A three level surrogate modeling strategy is suggested to reduce the computation time for the solution of optimization problems with polymorphic uncertain parameters defined as intervals and random variables. According to the four level algorithm in Fig. 2, the following surrogate models are introduced:

  • Level 1 surrogate model for the deterministic simulation

  • Level 2 surrogate model for the stochastic analysis

  • Level 3 surrogate model for both, the interval and the stochastic analyses

First, a

Verification of the multilevel ANN surrogate modeling approach

The proposed multilevel ANN surrogate modeling approach is verified by a benchmark optimization problem given by the following objective functionmin:{μu(Z(d1,d2,A))},withZ(d1,d2,A)=tanh(d12d2A)(3(1d1)2ed12(d2+1)210(d15d13d25)ed12d2213e(d1+1)2d22), considering two interval design parameters d1 and d2 and an additional uncertain a priori parameter (random variable A), see Table 1. In this example, 10000 samples of the random variable A are used to evaluate the

Conclusions

In this paper, a multilevel neural network based surrogate modeling strategy has been presented to solve optimization problems in structural mechanics considering stochastic and interval parameters. A first artificial neural network is trained to approximate the deterministic finite element analysis, which requires most of the numerical costs within the presented approach. Based on this neural network, a second neural network is created to replace the stochastic analysis (Monte Carlo

Declaration of Competing Interest

We wish to confirm that there are no known conflicts of interest associated with this publication and there has been no significant financial support for this work that could have influenced its outcome.

Acknowledgements

This research is funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) – Project number: 312921814 within Subproject 6 of the Priority Programme SPP 1886 “Polymorphic uncertainty modelling for the numerical design of structures”.

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