Gradient-based Solver Research Articles

Optimization of functionally graded materials to make stress concentration vanish in a plate with circular hole

This paper is devoted to the minimization of the stress concentration factor in infinite plates with circular hole made of functionally graded materials and subjected to a far-field uniform uniaxial tension. Despite the vast literature on the versatility of these materials, the novelty of the results is that the material distribution is not limited to prefixed laws, as in many works available in the literature. Instead, it is assumed to be an unknown piecewise constant function, thus aiming to derive the material distribution by exploiting, at best, the inhomogeneity concept associated with functionally graded materials. After a brief review of the governing equations, the motivation, the statement and the mathematical formulation of the optimization problem are given under the hypothesis of axisymmetric material distribution. Still, the problem could not be solved analytically, therefore a direct transcription approach by the aid of finite difference method has been followed to convert it into a nonlinear programming problem, whose solution has been obtained numerically by dedicated gradient-based solvers. Numerical optimal solutions are reported in graphical forms, thoroughly discussed and validated by means of the finite element method. The developed numerical approach yields a material inhomogeneity obeying a sigmoid-like function and a uniform hoop stress along the radial direction, thus making the stress concentration factor at the rim of the circular hole vanish.

Gradient-based wind farm layout optimization with inclusion and exclusion zones

Abstract. Wind farm layout optimization is usually subjected to boundary constraints of irregular shapes. The analytical expressions of these shapes are rarely available, and, consequently, it can be challenging to include them in the mathematical formulation of the problem. This paper presents a new methodology to integrate multiple disconnected and irregular domain boundaries in wind farm layout optimization problems. The method relies on the analytical gradients of the distances between wind turbine locations and boundaries, which are represented by polygons. This parameterized representation of boundary locations allows for a continuous optimization formulation. A limitation of the method, if combined with gradient-based solvers, is that wind turbines are placed within the nearest polygons when the optimization is started in order to satisfy the boundary constraints; thus the allocation of wind turbines per polygon is highly dependent on the initial guess. To overcome this and improve the quality of the solutions, two independent strategies are proposed. A case study is presented to demonstrate the applicability of the method and the proposed strategies. In this study, a wind farm layout is optimized in order to maximize the annual energy production (AEP) in a non-uniform wind resource site. The problem is constrained by the minimum distance between wind turbines and five irregular polygon boundaries, defined as inclusion zones. Initial guesses are used to instantiate the optimization problem, which is solved following three independent approaches: (1) a baseline approach that uses a gradient-based solver; (2) approach 1 combined with the relaxation of the boundaries, which allows for a better design space exploration; and (3) the application of a heuristic algorithm, “smart-start”, prior to the gradient-based optimization, improving the allocation of wind turbines within the inclusion polygons based on the potential wind resource and the available area. The results show that the relaxation of boundaries combined with a gradient-based solver achieves on average +10.2 % of AEP over the baseline, whilst the smart-start algorithm, combined with a gradient-based solver, finds on average +20.5 % of AEP with respect to the baseline and +9.4 % of AEP with respect to the relaxation strategy.

Identification of the tip mass parameters in a beam-tip mass system using response surface methodology

Abstract In this study, a response surface based approach is introduced to determine the physical parameters of the tip mass of a beam – tip mass system, such as mass, mass moment of inertia and coordinates of the centre of gravity with respect to the beam end point. To this end, first, a difference function was formulated based on the differences between the peak frequencies and peak amplitudes of the experimental and analytical frequency response functions. Later, observation points were established in the design space using orthogonal arrays, and a response surface was developed using the difference function values at these points. Next, the tip mass parameters were determined by minimizing the response surface with genetic algorithm and particle swarm optimization as well as fmincon, a gradient-based solver of the Matlab program. For comparison purposes, those parameters were obtained by also direct minimization of the difference function with the same algorithms. It was concluded that the tip mass parameters were successfully determined within reasonable error limits by the response surface method with less computational burden. Finally, the effect of design space width on the response surface quality is demonstrated numerically.

Gradient-Based Local Causal Structure Learning.

Finding the causal structure from a set of variables given observational data is a crucial task in many scientific areas. Most algorithms focus on discovering the global causal graph but few efforts have been made toward the local causal structure (LCS), which is of wide practical significance and easier to obtain. LCS learning faces the challenges of neighborhood determination and edge orientation. Available LCS algorithms build on conditional independence (CI) tests, they suffer the poor accuracy due to noises, various data generation mechanisms, and small-size samples of real-world applications, where CI tests do not work. In addition, they can only find the Markov equivalence class, leaving some edges undirected. In this article, we propose a GradieNt-based LCS learning approach (GraN-LCS) to determine neighbors and orient edges simultaneously in a gradient-descent way, and, thus, to explore LCS more accurately. GraN-LCS formulates the causal graph search as minimizing an acyclicity regularized score function, which can be optimized by efficient gradient-based solvers. GraN-LCS constructs a multilayer perceptron (MLP) to simultaneously fit all other variables with respect to a target variable and defines an acyclicity-constrained local recovery loss to promote the exploration of local graphs and to find out direct causes and effects of the target variable. To improve the efficacy, it applies preliminary neighborhood selection (PNS) to sketch the raw causal structure and further incorporates an l1 -norm-based feature selection on the first layer of MLP to reduce the scale of candidate variables and to pursue sparse weight matrix. GraN-LCS finally outputs LCS based on the sparse weighted adjacency matrix learned from MLPs. We conduct experiments on both synthetic and real-world datasets and verify its efficacy by comparing against state-of-the-art baselines. A detailed ablation study investigates the impact of key components of GraN-LCS and the results prove their contribution.

Towards Scalable Multi-View Reconstruction of Geometry and Materials.

In this paper, we propose a novel method for joint recovery of camera pose, object geometry and spatially-varying Bidirectional Reflectance Distribution Function (svBRDF) of 3D scenes that exceed object-scale and hence cannot be captured with stationary light stages. The input are high-resolution RGB-D images captured by a mobile, hand-held capture system with point lights for active illumination. Compared to previous works that jointly estimate geometry and materials from a hand-held scanner, we formulate this problem using a single objective function that can be minimized using off-the-shelf gradient-based solvers. To facilitate scalability to large numbers of observation views and optimization variables, we introduce a distributed optimization algorithm that reconstructs 2.5D keyframe-based representations of the scene. A novel multi-view consistency regularizer effectively synchronizes neighboring keyframes such that the local optimization results allow for seamless integration into a globally consistent 3D model. We provide a study on the importance of each component in our formulation and show that our method compares favorably to baselines. We further demonstrate that our method accurately reconstructs various objects and materials and allows for expansion to spatially larger scenes. We believe that this work represents a significant step towards making geometry and material estimation from hand-held scanners scalable.

On the Use of Gradient-Based Solver and Deep Learning Approach in Hierarchical Control: Application to Grand Refrigerators

This paper extends the work that has been recently studied on hierarchical control proposed by Alamir et al. (2017). This framework is designed to control interconnecting subsystems such as cryogenic processes or power plants. Based on the previous study, Pham et al. (2022) have shown that handling constraints and non-linearities could dispute the real-time feasibility of the approach. In order to reduce the computation time of the nonlinear model predictive controllers (NMPCs) of the local subsystems, two successful directions are investigated and combined, namely truncated fast gradient based NMPC approach and deep-neural-network-based approach. It is also shown that by doing so, the control updating period can be significantly reduced and the closed-loop performance is greatly improved. This paper can therefore be considered as a concrete implementation and validation of some key ideas in the design of real-time distributed NMPCs. All concepts are validated using the realistic and challenging example of a real cryogenic refrigerator.

In-plane Strain Analysis by Correlating Geometry and Visual Data Through a Gradient-Based Surface Reconstruction.

Abnormalities in tissue can be detected and analyzed by evaluating mechanical properties, such as strain and stiffness. While current sensor systems are effective in measuring longitudinal properties perpendicular to the measurement sensor, identifying in-plane deformation remains a significant challenge. To address this issue, this paper presents a novel method for reconstructing in-plane deformation of observed tissue surfaces using a fringe projection sensor specifically designed for measuring tissue deformations. The method employs the latest techniques from computer vision, such as differentiable rendering, to formulate the in-plane reconstruction as a differentiable optimization problem. This enables the use of gradient-based solvers for an efficient and effective optimization of the problem optimum. Depth information and image information are combined using landmark correspondences between the respective image observations of the undeformed and deformed scenes. By comparing the reconstructed pre- and post-deformation geometry, the in-plane deformation can be revealed through the analysis of relative variations between the corresponding models' geometries. The proposed reconstruction pipeline is validated on an experimental setup, and the potential for intraoperative applications is discussed.

Multimodal 4DVarNets for the Reconstruction of Sea Surface Dynamics From SST-SSH Synergies

The space-time reconstruction of sea surface dynamics from satellite observations is a challenging inverse problem due to the associated irregular sampling. Satellite altimetry provides a direct observation of the sea surface height (SSH), which relates to the divergence-free component of sea surface currents. The associated sampling pattern prevents operational schemes from retrieving fine-scale dynamics, typically below 10 days. By contrast, other satellite sensors provide higher-resolution observations of sea surface tracers such as sea surface temperature (SST). Multimodal inversion schemes then arise as appealing approaches. Though theoretical evidence supports the existence of an explicit relationship between sea surface temperature and sea surface dynamics under specific dynamical regimes, the generalization to the variety of upper ocean dynamical regimes is complex. Here, we investigate this issue from a physics-informed learning perspective. We introduce a trainable multimodal inversion scheme for the reconstruction of sea surface dynamics from multi-source satellite-derived observations, namely satellite-derived SSH and SST data. The proposed multimodal 4DVarNet schemes combine a variational formulation involving trainable observation and <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">a priori</i> terms with a trainable gradient-based solver. An observing system simulation experiment for a Gulf Stream region supports the relevance of our approach compared with state-of-the-art schemes. We report a relative improvement greater than 60% compared with the operational altimetry product in terms of root mean square error and resolved space-time scales. We discuss further the potential and the limitations of the proposed approach for the reconstruction and forecasting of geophysical dynamics from irregularly-sampled satellite observations.

Buckling optimization of variable-stiffness composites with multiple cutouts considering manufacturing constraints

The progress of the automatic fiber placement (AFP) technique makes it viable to fabricate variable-stiffness (VS) composites with curvilinear fiber paths and promotes the rise of the VS composite design. However, traditional representation methods of fiber paths are not flexible for composites with cutouts and are also inconvenient to cooperate with the gradient-based solver. In this study, a parameterized angle variable scheme (PAVS) by the compactly supported radial basis functions (CS-RBFs) is proposed to represent the continuous fiber paths so that manufacturing constraints including minimal turning radius and gap/overlap can be conveniently related to the curl operation and divergence of the fiber angle vector field. A level set representation method of fiber paths is also provided for comparison. Two buckling optimization frameworks of the VS composite considering manufacturing constraints are then proposed based on these two representation methods. Numerical examples of two square plates with cutouts are conducted. Different numbers of support points, support radii, manufacturing constraints, and initial designs are investigated. Results indicate that the proposed PAVS needs few support points and a small support radius. It can also set initial design easily and describe the manufacturing constraints directly and quantitively.

Optimally Safe Tank Changeover Operation Using a Smooth Optimization Formulation.

Tank changeover is a routine process in industry for placing fuel tanks into or out of service. The operation must use inert gas to avoid the flammability zone. However, inert gas consumption should be minimized for economic reasons. This requires dynamic modeling and optimization of the process, as addressed in the present work. A new dynamic optimization problem for minimizing the inert gas consumption, while ensuring fire safety is proposed. As part of the problem constraints, the flammability zone is characterized by disjunctive constraints, which are then converted to a new simple, nonsmooth formula, removing the need for data regression. This together with the multi-mode flow equations in the model leads to a nonsmooth dynamic optimization problem. To enable reliable solution by gradient-based solvers, the problem is reformulated to a smooth one using sigmoid functions. Case studies of methane tank purging and filling operations demonstrate that the proposed approach is able to minimize the inert consumption by providing optimal trajectories of the tank inlet and outlet flow rates, while ensuring the operation remains outside the flammability zone. It is shown that the proposed dynamic optimization can yield significant economic benefits as it reduces the nitrogen consumption by about two-third in one of the examples solved.

Development of a Location Optimization Method for Embedded Components

Embedded components can be found in structures such as multi-material additively manufactured structures and sensors embedded for health monitoring. The embedded components often have different materials from the base structure. Variation of Young’s modulus from the base structure to the embedded components can lead to variation in stress values. Variation of material strengths leads to the load bearing capacity of the component being different from the base structure. The location of embedded components has a significant influence on the stress in these components. An appropriate choice of component location could significantly reduce its stress. In this paper, a location optimization method is developed to reduce the stress in components embedded inside the structures. The sensitivity of maximum von Mises stress around the component with respect to its placement is calculated through the adjoint method. A gradient-based solver is applied to solve the optimization problem. A cantilever beam with one component was used to demonstrate the feasibility of the method. The method was applied to two case studies and significant reduction in the maximum von Mises stress of component was achieved.

Noise Robust High-Speed Motion Compensation for ISAR Imaging Based on Parametric Minimum Entropy Optimization

When a target is moving at high-speed, its high-resolution range profile (HRRP) will be stretched by the high-order phase error caused by the high velocity. In this case, the inverse synthetic aperture radar (ISAR) image would be seriously blurred. To obtain a well-focused ISAR image, the phase error induced by target velocity should be compensated. This article exploits the variation continuity of a high-speed moving target’s velocity and proposes a noise-robust high-speed motion compensation algorithm for ISAR imaging. The target’s velocity within a coherent processing interval (CPI) is modeled as a high-order polynomial based on which a parametric high-speed motion compensation signal model is developed. The entropy of the ISAR image after high-speed motion compensation is treated as an evaluation metric, and a parametric minimum entropy optimization model is established to estimate the velocity and compensate it simultaneously. A gradient-based solver of this optimization is then adopted to iteratively find the optimal solution. Finally, the high-order phase error caused by the target’s high-speed motion can be iteratively compensated, and a well-focused ISAR image can be obtained. Extensive simulation experiments have verified the noise robustness and effectiveness of the proposed algorithm.

Particle Swarm Optimization for Dynamic Risk-Aware Path Following for Autonomous Ships

This work presents the use of particle swarm optimization (PSO) for dynamic risk-aware path following, or waypoint re-planning during autonomous surface navigation in a maritime environment with polygonal grounding obstacles. Although recent research on control and local or global path planning for maritime autonomous surface ships (MASS) is considerable, few deal with the concept of risk during dynamic path following along a preplanned path. The proposed method introduces risk-based terms in the PSO fitness function for dynamic (online) adjustment or re-planning of intermediate waypoints during path following of a pre-planned route, where the control of the vessel in this work is left to a standard line-of-sight PID controller. Moreover, the results are compared to the performance of an analogous implementation of risk-aware model predictive control (MPC) using a gradient-based solver. The suggested method yields adequate performance similar to that of the MPC algorithm. Ultimately, the PSO approach may in future works allow for more complex and dynamic risk-aware behavior related to e.g. weather conditions implemented through lookup tables, or discrete machinery modes that are non-smooth.

Optimizing global injectivity for constrained parameterization

Injective parameterizations of triangulated meshes are critical across applications but remain challenging to compute. Existing algorithms to find injectivity either require initialization from an injective starting state, which is currently only possible without positional constraints, or else can only prevent triangle inversion, which is insufficient to ensure injectivity. Here we present, to our knowledge, the first algorithm for recovering a globally injective parameterization from an arbitrary non-injective initial mesh subject to stationary constraints. These initial meshes can be inverted, wound about interior vertices and/or overlapping. Our algorithm in turn enables globally injective mapping for meshes with arbitrary positional constraints. Our key contribution is a new energy, called smooth excess area (SEA), that measures non-injectivity in a map. This energy is well-defined across both injective and non-injective maps and is smooth almost everywhere, making it readily minimizable using standard gradient-based solvers starting from a non-injective initial state. Importantly, we show that maps minimizing SEA are guaranteed to be locally injective and almost globally injective, in the sense that the overlapping area can be made arbitrarily small. Analyzing SEA's behavior over a new benchmark set designed to test injective mapping, we find that optimizing SEA successfully recovers globally injective maps for 85% of the benchmark and obtains locally injective maps for 90%. In contrast, state-of-the-art methods for removing triangle inversion obtain locally injective maps for less than 6% of the benchmark, and achieve global injectivity (largely by chance as prior methods are not designed to recover it) on less than 4%.

Simultaneous Optimization of an IGCC Process with Equation-Oriented Methods

Simultaneous optimization of process flowsheets using equation-oriented (EO) methods is a promising alternative to simulation studies. With the use of EO methods, simultaneous optimization of all process units including heat integration can be performed to achieve goals such as maximizing profit or minimizing operating costs. The use of EO methods required the development of new process models and equations of state (EOS), as algorithms and some formulations used in sequential methods do not work in an EO environment. Advancements in EO methods, along with the growth of gradient-based solvers, have enabled large-scale simultaneous flowsheet optimization. However, simultaneous process optimization still faces challenges as large problems, especially non-linear ones, may not achieve convergence without good initial values. A study on the simultaneous optimization of the integrated gasification and combined cycle (IGCC) process using EO methods was conducted as it is a combination of a cryogenic air separation unit (ASU), a coal gasifier and power generation units consisting of a combustor, a gas turbine, a heat recovery steam generator (HRSG) and a steam turbine. The purity of the oxygen product fed to the gasifier was often considered to be fixed in prior IGCC studies. With the ASU model and gasifier model used, the influence of oxygen purity and flowrate fed to the gasifier can be studied. The thermal efficiency of the IGCC process was maximized in GAMS using full-order EO process models. After the optimization, the thermal efficiency increased from 27.7 % in the base case to 36.7 %, with a notable change in process parameters, showing that the solver did not terminate close to the starting solution. Results showed that lowering the oxygen product purity from 95 % to 58.6 % increased the thermal efficiency of the IGCC process as the power required by the compressor in the ASU was reduced.

Goal-oriented mesh adaptivity for inverse problems in linear micromorphic elasticity

In this work, we extend goal-oriented mesh adaptivity to parameter identification for a class of linear micromorphic elasticity problems. Starting from a compact formulation in our previous work (Ju and Mahnkhen, 2017), we propose a two-level optimization framework based on goal-oriented error estimation. By means of a sensitivity analysis of the generalized constitutive relations, we establish a gradient-based solver for the inverse problem, where parameters are optimized within an inner optimization loop for a given mesh. Exact error representations are derived from a Lagrange method, aiming at a quantity of interest as a user-defined functional of the parameters. By using a patch recovery technique for enhanced solutions, a computable error estimator is presented and used to drive an adaptive refinement algorithm, which forms an outer optimization loop. For a numerical study, we investigate the performance of the resulting adaptive procedure in case of perfect, incomplete and perturbed data. The results confirm the effectiveness of the proposed adaptive procedure.

Topology optimization for fail-safe designs using moving morphable components as a representation of damage

Designs obtained with topology optimization (TO) are usually not safe against damage. In this paper, density-based TO is combined with a moving morphable component (MMC) representation of structural damage in an optimization problem for fail-safe designs. Damage is inflicted on the structure by an MMC which removes material, and the goal of the design problem is to minimize the compliance for the worst possible damage. The worst damage is sought by optimizing the position of the MMC to maximize the compliance for a given design. This non-convex problem is treated using a gradient-based solver by initializing the MMC at multiple locations and taking the maximum of the compliances obtained. The use of MMCs to model damage gives a finite element-mesh-independent method, and by allowing the components to move rather than remain at fixed locations, more robust structures are obtained. Numerical examples show that the proposed method can produce fail-safe designs with reasonable computational cost.

Robust implementation of generative modeling with parametrized quantum circuits

Although the performance of hybrid quantum-classical algorithms is highly dependent on the selection of the classical optimizer and the circuit ansätze (Benedetti et al, npj Quantum Inf 5:45, 2019; Hamilton et al, 2018; Zhu et al, 2018), a robust and thorough assessment on-hardware of such features has been missing to date. From the optimizer perspective, the primary challenge lies in the solver’s stochastic nature, and their significant variance over the random initialization. Therefore, a robust comparison requires one to perform several training curves for each solver before one can reach conclusions about their typical performance. Since each of the training curves requires the execution of thousands of quantum circuits in the quantum computer, such a robust study remained a steep challenge for most hybrid platforms available today. Here, we leverage on Rigetti’s Quantum Cloud Services (QCS™) to overcome this implementation barrier, and we study the on-hardware performance of the data-driven quantum circuit learning (DDQCL) for three different state-of-the-art classical solvers, and on two-different circuit ansätze associated to different entangling connectivity graphs for the same task. Additionally, we assess the gains in performance from varying circuit depths. To evaluate the typical performance associated with each of these settings in this benchmark study, we use at least five independent runs of DDQCL towards the generation of quantum generative models capable of capturing the patterns of the canonical Bars and Stripes dataset. In this experimental benchmarking, the gradient-free optimization algorithms show an outstanding performance compared to the gradient-based solver. In particular, one of them had better performance when handling the unavoidable noisy objective function to be minimized under experimental conditions.

Evaluating gantry crane-way pavement performance: An inverse approach

Gantry-crane pavements and foundations are significant assets within intermodal facilities. The performance of these pavements, which are subjected to highly-variable loads, is critical to safe operations. Traffic interruptions and costs associated with maintaining and rehabilitating distressed or failed pavements in associated areas are of particular importance. The purpose of this study was to evaluate structural behavior and improve design procedures for gantry crane way pavement used at intermodal facilities by assessing interactions among pavements, subgrades, and operational loading conditions. The performance of the gantry crane pavements and foundations was assessed using a finite-element (FE) model, while pavement structural response to a crane load was measured using strain gages installed in the field. Measuring material properties using in-situ/laboratory tests was not possible in this work due to the limited resources, restricted access to intermodal facilities, and tight schedule of rail freight transport(i.e. tight schedule of cranes). To verify the materials properties and ensure a consistent behavior of the developed FE model with respect to the field measurements, an inverse design approach was implemented. Because of the significant cost of FE simulation, the gradient-based solver used in this study is based on a trust region algorithm. The verified model was used to predict the critical responses of Portland cement concrete (PCC) layer, base course, and subgrade soil. These parameters were then used to conduct a pavement fatigue damage analysis, parametric analyses of material strength and slab geometry were carried out based on Model Code, and resulting fatigue-life recommendations for improved designs were made. The developed FE model along with the inverse approach create a framework that can be used in further health monitoring problems when a specific parameter cannot be directly observed.

Lights and shadows in Evolutionary Deep Learning: Taxonomy, critical methodological analysis, cases of study, learned lessons, recommendations and challenges

Much has been said about the fusion of bio-inspired optimization algorithms and Deep Learning models for several purposes: from the discovery of network topologies and hyperparametric configurations with improved performance for a given task, to the optimization of the model’s parameters as a replacement for gradient-based solvers. Indeed, the literature is rich in proposals showcasing the application of assorted nature-inspired approaches for these tasks. In this work we comprehensively review and critically examine contributions made so far based on three axes, each addressing a fundamental question in this research avenue: (a) optimization and taxonomy (Why?), including a historical perspective, definitions of optimization problems in Deep Learning, and a taxonomy associated with an in-depth analysis of the literature, (b) critical methodological analysis (How?), which together with two case studies, allows us to address learned lessons and recommendations for good practices following the analysis of the literature, and (c) challenges and new directions of research (What can be done, and what for?). In summary, three axes – optimization and taxonomy, critical analysis, and challenges – which outline a complete vision of a merger of two technologies drawing up an exciting future for this area of fusion research.