Physics-based Surrogate Optimization of Francis Turbine Runner Blades, Using Mesh Adaptive Direct Search and Evolutionary Algorithms

This article, written by JPT Technology Editor Chris Carpenter, contains highlights of paper SPE 193895, “Multilevel Strategies and Geological Parameterizations for History Matching Complex Reservoir Models,” by Yimin Liu, SPE, and Louis J. Durlofsky, SPE, Stanford University, prepared for the 2019 SPE Reservoir Simulation Conference, Galveston, Texas, 10–11 April. The paper has not been peer reviewed. The complete paper explores the use of multilevel derivative-free optimization for history matching, with model properties described using principal component analysis (PCA) -based parameterization techniques. The parameterizations applied are optimization-based PCA (O-PCA) and convolutional-neural-network-based PCA (CNN-PCA). Mesh adaptive direct search (MADS), a pattern search method that parallelizes naturally, is used for the optimizations required to generate posterior models. The use of PCA-based parameterization reduces considerably the number of variables that must be determined during history matching, but the optimization problem can still be computationally demanding. The multilevel strategy addresses this issue by reducing the number of simulations that must be performed at each MADS iteration. History-matching results demonstrate that substantial uncertainty reduction is achieved in all cases considered and that the multilevel strategy is effective in reducing the number of simulations required. Overview Parameterization has two key advantages in history matching: Far fewer parameters must be determined, which simplifies greatly the minimization in some cases. Posterior (history-matched) geomodels will be consistent with the geological scenario or training image for which the parameterization was constructed. PCA is the foundation for many geological-dimension-reduction parameterization methods. PCA is based on the eigen-decomposition of the prior geomodel covariance matrix and, thus, honors only two-point spatial statistics. History-matching algorithms largely can be classified as either optimization-based or ensemble-based procedures. Optimization-based algorithms generate one history-matched model at a time, while ensemble-based algorithms update a large set of models simultaneously. The resulting ensemble of posterior models is used to quantify uncertainty. In this study, only optimization-based history matching is considered, though the parameterizations applied also can be used with ensemble-based methods. Optimization-based history-matching algorithms can be classified further into gradient-based and derivative-free optimization (DFO) methods. Gradient-based algorithms often are combined with adjoint procedures that provide the required gradient efficiently. Unlike adjoint-based procedures, DFO algorithms (which are considered in this work) are nonintrusive. They treat the reservoir simulator as a so-called “black box,” and, as a result, they are much easier to implement and can be applied with any simulator. In this study, MADS, a pattern-search DFO, is applied for history matching. In pattern-search (or stencil-based) algorithms, the number of simulations required at each iteration scales with the number of parameters to be determined. Because the latter typically is much smaller than the number of grid-blocks in the model, geological parameterization leads to large computational savings when a pattern search, or other DFO, is used for history matching. The performance of DFO for history matching with non-Gaussian models, a primary topic in the complete paper, has not been studied extensively. The authors devise a multilevel strategy, applicable for PCA-based parameterization methods, to further reduce the computational cost of history matching using DFO. A multilevel strategy lends itself naturally to PCA-based parameterizations because the PCA representation can be ordered from the most-important to the least-important principal component. Thus, PCA parameters can be determined sequentially, level by level, in groups, rather than all at once.

A mesh adaptive direct search algorithm for multiobjective optimization

Combined heat and power economic dispatch by mesh adaptive direct search algorithm

Inversion of surface wave phase velocities not only suffers from computational difficulties associated with being trapped by local minima for most local-search methods but also undergoes the high computational effort for most global optimization techniques due to its high nonlinearity and to its multimodality. In order to effectively overcome the above described difficulties, we proposed a new surface wave inversion scheme based on the Mesh Adaptive Direct Search (MADS) algorithm, a commonly used pattern recognition algorithm. The calculation efficiency and stability of the proposed procedure are tested on a four-layer synthetic model and a real-world example. Results from both synthetic and real field data demonstrate that the MADS algorithm applied to surface wave inversion should be considered good not only in terms of accuracy but also in terms of computation effort due to its global and fast convergence in the final stage of search.

Summary In this study, we explore using multilevel derivative-free optimization (DFO) for history matching, with model properties described using principal-component-analysis (PCA) -based parameterization techniques. The parameterizations applied in this work are optimization-based PCA (O-PCA) and convolutional-neural-network (CNN) -based PCA (CNN-PCA). The latter, which derives from recent developments in deep learning, is able to accurately represent models characterized by multipoint spatial statistics. Mesh adaptive direct search (MADS), a pattern-search method that parallelizes naturally, is applied for the optimizations required to generate posterior (history-matched) models. The use of PCA-based parameterization considerably reduces the number of variables that must be determined during history matching (because the dimension of the parameterization is much smaller than the number of gridblocks in the model), but the optimization problem can still be computationally demanding. The multilevel strategy introduced here addresses this issue by reducing the number of simulations that must be performed at each MADS iteration. Specifically, the PCA coefficients (which are the optimization variables after parameterization) are determined in groups, at multiple levels, rather than all at once. Numerical results are presented for 2D cases, involving channelized systems (with binary and bimodal permeability distributions) and a deltaic-fan system using O-PCA and CNN-PCA parameterizations. O-PCA is effective when sufficient conditioning (hard) data are available, but it can lead to geomodels that are inconsistent with the training image when these data are scarce or nonexistent. CNN-PCA, by contrast, can provide accurate geomodels that contain realistic features even in the absence of hard data. History-matching results demonstrate that substantial uncertainty reduction is achieved in all cases considered, and that the multilevel strategy is effective in reducing the number of simulations required. It is important to note that the parameterizations discussed here can be used with a wide range of history-matching procedures (including ensemble methods), and that other derivative-free optimization methods can be readily applied within the multilevel framework.

The context of this research is multiobjective optimization where conflicting objectives are present. In this work, these objectives are only available as the outputs of a blackbox for which no derivative information is available. This work proposes a new extension of the mesh adaptive direct search (MADS) algorithm to multiobjective derivative-free optimization with bound constraints. This method does not aggregate objectives and keeps a list of non dominated points which converges to a (local) Pareto set as long as the algorithm unfolds. As in the single-objective optimization MADS algorithm, this method is built around a search step and a poll step. Under classical direct search assumptions, it is proved that the so-called DMulti-MADS algorithm generates multiple subsequences of iterates which converge to a set of local Pareto stationary points. Finally, computational experiments suggest that this approach is competitive compared to the state-of-the-art algorithms for multiobjective blackbox optimization.

Black box operation optimization of basic oxygen furnace steelmaking process with derivative free optimization algorithm

Optimization solvers include many options that allow users to control algorithmic aspects that may have a considerable impact on solver performance. Tuning solver options is often necessary to reduce execution time and improve solution quality. Previous studies of solver tuning techniques have focused on mixed-integer linear programming and local nonlinear programming solvers. In this paper, we investigate the potential of tuning a global optimization solver for nonlinear and mixed-integer nonlinear programming problems. In particular, derivative-free optimization (DFO) algorithms are used to find optimal values for options of the global optimization solver BARON. A set of 126 problems from the GLOBALLib and MINLPLib collections are utilized in a computational study from which we conclude that tuning options can improve the default performance of BARON for individual problems and an entire library. Additionally, we present a systematic comparison of 27 DFO solvers in terms of their ability to improve the performance of the global solver. We find that several DFO implementations are much better than others in terms of finding optimal tuning parameters.

Bistable mechanical systems exhibit two stable configurations where the elastic energy is locally minimized. To realize such systems, origami techniques have been proposed as a versatile platform to design deployable structures with both compact and functional stable states. Conceptually, a bistable origami motif is composed of two-dimensional surfaces connected by one-dimensional fold lines. This leads to stable configurations exhibiting zero-energy local minima. Physically, origami-inspired structures are three-dimensional, comprising facets and hinges fabricated in a distinct stable state where residual stresses are minimized. This leads to the dominance of one stable state over the other. To improve mechanical performance, one can solve the constrained optimization problem of maximizing the bistability of origami structures, defined as the amount of elastic energy required to switch between stable states, while ensuring materials used for the facets and hinges remain within their elastic regime. In this study, the Mesh Adaptive Direct Search (Mads) algorithm, a blackbox optimization technique, is used to solve the constrained optimization problem. The bistable waterbomb-base origami motif is selected as a case-study to present the methodology. The elastic energy of this origami pattern under deployment is calculated via Finite Element simulations which serve as the blackbox in the Mads optimization loop. To validate the results, optimized waterbomb-base geometries are built via Fused Filament Fabrication and their response under loading is characterized experimentally on a Uniaxial Test Machine. Ultimately, our method offers a general framework for optimizing bistability in mechanical systems, presenting opportunities for advancement across various engineering applications.

The calibration of hydrological models is formulated as a blackbox optimization problem where the only information available is the objective function value. Distributed hydrological models are generally computationally intensive, and their calibration may require several hours or days which can be an issue for many operational contexts. Different optimization algorithms have been developed over the years and exhibit different strengths when applied to the calibration of computationally intensive hydrological models. This paper shows how the dynamically dimensioned search (DDS) and the mesh adaptive direct search (MADS) algorithms can be combined to significantly reduce the computational time of calibrating distributed hydrological models while ensuring robustness and stability regarding the final objective function values. Five transitional features are described to adequately merge both algorithms. The hybrid approach is applied to the distributed and computationally intensive HYDROTEL model on three different river basins located in Québec (Canada).

It is proposed a first step to the model-based closed-loop control of a separated flow. In such situations, fluid mechanics phenomena are highly nonlinear and can be represented by means of the Navier-Stokes equations. However, such a model still rises difficult issues for control practice. This paper proposes an alternative, bilinear and delayed model, the accuracy of which is studied. The identification technique combines least-square technique with a Mesh Adaptive Direct Search (MADS) algorithm. The main feature of the model is state dependent structure of input delay.

Designing icephobic surfaces to delay ice formation is crucial for applications like aviation safety and cryopreservation. While classical nucleation theory (CNT) provides a thermodynamic foundation, real-world stochastic effects and complex wetting states make designing these surfaces difficult. The performance of icephobic surfaces is influenced by surface structure. However, finding the best designs for specific environmental conditions is still a challenge. In this work, we introduce a hybrid approach that combines experiments, CNT, and blackbox optimization to predict and optimize ice nucleation time on micropatterned surfaces. Cylindrical SU-8 micropillar arrays with different heights and spacings were created, and the apparent contact angles and freezing delay times were measured at -10 °C and -20 °C. An analytical model was developed to describe the wetting states between the Wenzel and Cassie-Baxter regimes, optimizing its parameters using the Mesh adaptive direct search (MADS) algorithm. This approach allows us to estimate contact angles and ice nucleation times for any surface geometry within the studied design space. The predicted contact angles matched experimental results with a mean absolute percentage error (MAPE) of 2.09% (R2 = 0.92) and the approximate nucleation times showed a MAPE of 27.3% (R2 = 0.75). Our method also identified the best micropillar geometries that maximize freezing delay and was validated with an independent data set, showing strong predictive ability. This work emphasizes the benefit of combining physics-based models with data-driven optimization for rapid design of icephobic surfaces.

In the design of optical metamaterials, some optimization problems require launching a numerical simulation. The Mesh Adaptive Direct Search algorithm is designed for such problems. The MADS algorithm does not require any derivative information about the problem being optimized, and no continuity or differentiability assumptions are made by MADS on the functions defining the simulation. A detailed discussion of the method is provided in the second section of the chapter, followed by a discussion of the NOMAD implementation of the method and its features. The final section of the chapter lists three instances of combining NOMAD with Finite-Difference Time-Domain electromagnetic simulations to tailor the broadband spectral response and near-field interactions of Split Ring Resonator metamaterials.

In this work we present an approach to determine geologically plausible reservoir models that are consistent with well-log data and multimodal observations (e.g., production and seismic data). The methodology is based on optimization techniques that are not invasive to the simulators used to reproduce observations. The geologically plausible reservoir models obtained facilitate reliable probabilistic production forecasts and lead to more robust optimization and decision-making support tools. The methodology approaches model inversion as an optimization problem where the cost function is a measure of the discrepancy between simulated responses and measurement data. Reservoir model uncertainty quantification is addressed by means of multi-start optimization, i.e., the use of local optimization methods with multiple initial guesses. Parameter reduction techniques allow a more effective and efficient global exploration of the optimization space. Geological realism is incorporated in the approach through multiple-point statistics. The set of reservoir models, determined as the output of the multi-start optimization, is thereafter included in a framework for robust optimization under uncertainty (e.g., to determine the drilling location of a new well). The optimization algorithms used are noninvasive with respect to the simulators used in the workflow and are implemented on a distributed- computing cluster. Different aspects of this approach are illustrated by means of a comprehensive case example that uses well-log and production data. It is important to note that the use of simulator-noninvasive (black box, derivative-free) optimization algorithms reduces significantly the dependence of the methodology on the simulators. The optimization methods tested include Generalized Pattern Search, Mesh Adaptive Direct Search, Sequential Quadratic Programming (with numerical derivatives) and Particle Swarm Optimization, a population-based procedure. These algorithms may yield efficient performance in terms of wall-clock time when they are combined with parameter reduction techniques and distributed computing (as is the case in this work). When parallel computing resources are scarce or nonexistent one may resort to methods suited to serial implementations (examples of these simulator-noninvasive algorithms are Hooke-Jeeves Direct Search and model-based approaches). The parameter reduction techniques studied here are based on Principal Component Analysis. In particular, the recently introduced Optimization-based Principal Component Analysis has been observed to deal with complex geological models satisfactorily. Uncertainty in future production is then incorporated in the robust optimization of well placement and of reactive well controls. The Ensemble Kalman Filter is widely used in industry for reservoir model uncertainty quantification. This method fails in some occasions (e.g., the ensemble collapses) for reasons that are not yet well understood. The multistart methodology proposed in this work can be an alternative to the Ensemble Kalman Filter in these problematic situations. Seismic data could be included in this methodology to mitigate uncertainty in probabilistic forecasts.

NDA Approval