Minimization Formulation Research Articles

Robust topology optimization of trusses with automatic grouping of cross-sections

The ground structure method is a topology optimization strategy that tends to generate complex structural solutions to structural optimization problems. One of the main causes of this complexity is the large number of different cross-sections present in the topologies optimized by this method, in which the solutions obtained often have dozens of bars with different cross-sections. This excessive diversity of cross-sections makes the solutions obtained impractical for the manufacture and assembly of real structures, which require standardized cross-sections for economic and constructive viability. Therefore, in order to improve the workflow during the design stage, we propose the development of a strategy that allows the number of cross-sectional areas available to be restricted during the optimization process. The proposed method acts independently of the number of members present in the initial structure and ensures that the optimized solution has a predefined number of cross-sections with equal areas. To do this, an initial optimization is carried out to obtain an ordered vector of optimized design variables with a higher tolerance. From this ordering, the groupings of members that will be associated with the same design variable are defined, which generates a new optimization problem with a reduced number of design variables. The material of the structure is again redistributed among the members and the new optimization problem is solved. Structures were optimized with a robust compliance minimization formulation that introduces uncertainty in the loading directions. This formulation naturally increases the complexity of the final structure compared to a nominal formulation. As a result, simpler topologies are obtained that benefit the construction process by reducing the number of elements with different cross-sections in the structure. Numerical examples are presented to demonstrate the efficiency of the strategy developed.

Expected Residual Minimization Formulation for Stochastic Absolute Value Equations

Expected Residual Minimization Formulation for Stochastic Absolute Value Equations

An improved peridynamics topology optimization formulation for compliance minimization

This work proposes an improved peridynamics density-based topology optimization framework for compliance minimization. One of the main advantages of using a peridynamics discretization relies in the fact that it provides a consistent regularization of classical continuum mechanics into a nonlocal continuum, thus containing an inherent length scale called the horizon. Furthermore, this reformulation allows for discontinuities and is highly suitable for treating fracture and crack propagation. Partial differential equations are rewritten as integrodifferential equations and its numerical implementation can be straightforwardly done using meshfree collocation, inheriting its advantages. In the optimization formulation, Solid Isotropic Material with Penalization (SIMP) is used as interpolation for the design variables. To improve the peridynamic formulation and to evaluate the objective function in a energetically consistent manner, surface correction is implemented. Moreover, a detailed sensitivity analysis reveals an analytical expression for the objective function derivatives, different from an expression commonly used in the literature, providing an important basis for gradient-based topology optimization with peridynamics. The proposed implementation is studied with two examples illustrating different characteristics of this framework. The analytical expression for the sensitivities is validated against a reference solution, providing an improvement over the referred expression in the literature. Also, the effect of using the surface correction is evidenced. An extensive analysis of the horizon size and sensitivity filter radius indicates that the current method is mesh-independent, i.e. a sensitivity filter is redundant since peridynamics intrinsically filters length scales with the horizon. Different optimization methods are also tested for uncracked and cracked structures, demonstrating the capabilities and robustness of the proposed framework.

SWIRL: The First Australian Operational Radar-Based 3D Wind Analysis System

Abstract In this paper, the first Australian operational radar-based three-dimensional (3D) wind analysis system named Synthetic Wind Information from Radar and Lidar (SWIRL) is described and evaluated. SWIRL employs a variational minimization formulation to combine results from four individual wind retrieval techniques of varied complexity to derive 3D winds in single-Doppler and multi-Doppler radar regions: a variational version of the traditional velocity azimuth display (VVAD) and double VAD (DVAD) techniques, a single-Doppler wind retrieval technique using optical flow horizontal wind proxies, and a multi-Doppler 3D wind retrieval technique. The SWIRL 3D wind components are evaluated against wind profiler observations and radar simulations using a very high-resolution (50 m) numerical simulation of a supercell thunderstorm. We find that SWIRL can retrieve very accurate horizontal winds, especially below 2-km height in the multi-Doppler regions, with mean absolute errors on wind speed and direction < 2 m s−1 and 10° on average and <2.5 m s−1 and 15°–20° 90% of the time. These errors do not increase noticeably with wind speed, highlighting the suitability of these retrieved winds to be used for damaging and destructive wind detection and nowcasting. The single-Doppler retrieval using optical flow is also found to provide reasonably accurate winds at these heights. The accurate retrieval of convective-scale updrafts and downdrafts, even using multi-Doppler information, is still a major challenge, with mean absolute errors of vertical velocity of about 50% on average. This can be attributed to the limitations of the current radar technology used operationally, imposing slow antenna speeds. Significance Statement Damaging and destructive winds have the potential to inflict significant damage to properties and assets and, tragically, result in loss of life. Efficient direction of emergency services to affected areas is essential for a prompt return to normal conditions. Wind farm operators require precise information on anticipated wind shifts to reduce the risk of energy grid failures. Strong winds also contribute to compound weather events, such as water ingress through hail-damaged roofs or structural damage to buildings caused by hailstones. The purpose of this work was to equip Australia with the first operational wind monitoring system, based on operational radar observations, to serve all these critical applications (and more).

A mixed finite element method for the forward problem of electrical impedance tomography with the shunt model

We propose a mixed finite element method for the forward problem of electrical impedance tomography with the shunt model. The method is based on a constrained minimization formulation in terms of the current density, where parts of the Lagrange multiplier are interpreted as the voltages on the electrodes. We justify the well-posedness of this continuous formulation, estimate the regularity of its solution, and use Raviart–Thomas elements for its discretization. The resulting mixed method is then proved to be stable, and converge especially with respect to the voltages on the electrodes. Simulation on a 2D model also gives consistent results.

Multi-objective branch-and-bound for the design-for-control of water distribution networks with global bounds

This study investigates a design-for-control (DfC) problem formulation for the joint minimization of pressure-induced leakage, maximization of resilience, and minimization of cost in water distribution networks (WDN). The DfC problem, which consists in simultaneously installing new valves and/or pipes and optimizing valve control settings, results in a challenging optimization problem belonging to the class of non-convex multi-objective mixed-integer non-linear programs (MOMINLP). Due to their complex mathematical structure, multi-objective WDN design-for-control problems have previously been solved using general-purpose evolutionary algorithms or local deterministic methods, which do not provide guarantees on the quality of the returned solutions. While branch-and-bound (BB) frameworks have been proposed to approximate the Pareto fronts of MOMINLPs with global bounds, they rely on the availability of attainable solutions which, for WDN design-for-control problems, can be hard to identify. Moreover, in the absence of general-purpose solvers, the performance of multi-objective BB implementations depends, for a given application, on the choice of adequate branching and lower bounding strategies. In this study, we investigate a multi-objective BB algorithm based on tailored branching, lower bounding and additional upper bounding strategies to efficiently approximate the Pareto front of WDN design-for-control problems with bounds of ϵ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\epsilon$$\\end{document}-non-dominance. The proposed algorithm is applied to the optimal DfC of two case study networks and is shown to outperform alternative solution methods.

Scalar auxiliary variable approach in iterative minimization formulation for saddle point search

Scalar auxiliary variable approach in iterative minimization formulation for saddle point search

Stress-Constrained Topology Optimization for Commercial Software: A Python Implementation for ABAQUS®

Topology optimization has evidenced its capacity to provide new optimal designs in many different disciplines. However, most novel methods are difficult to apply in commercial software, limiting their use in the academic field and hindering their application in the industry. This article presents a new open methodology for solving geometrically complex non-self-adjoint topology optimization problems, including stress-constrained and stress minimization formulations, using validated FEM commercial software. The methodology was validated by comparing the sensitivity analysis with the results obtained through finite differences and solving two benchmark problems with the following optimizers: Optimality Criteria, Method of Moving Asymptotes, Sequential Least-Squares Quadratic Programming (SLSQP), and Trust-constr optimization algorithms. The SLSQP and Trust-constr optimization algorithms obtained better results in stress-minimization problem statements than the methodology available in ABAQUS®. A Python implementation of this methodology is proposed, working in conjunction with the commercial software ABAQUS® 2023 to allow a straightforward application to new problems while benefiting from a graphic user interface and validated finite element solver.

Generating probability distributions on intervals and spheres: Convex decomposition

This work studies the convexity of a class of positive definite probability measures with density functionsρ=cλ∑n=0∞g(n)Cnλwλ, that are generated through the interval characteristic sequences. We discuss when such measures supported in an interval are convex and demonstrate how convexity is preserved under multiplication by certain multipliers and under the intertwining product. A key message to be delivered from this work is: any integrable function f on the interval with a polynomial expansion that has relatively fast absolute convergence, can be decomposed into a non-unique pair of positive convex interval probabilities (f+,f−)∈Eλ′×Eλ′ up to normalization, in the sense that f=f+−f−, where Eλ′ is a class of convex functions. Consequently not only the study of an interval distribution is reduced to the study of the properties of class Eλ′, but also the discontinuous probabilistic Galerkin schemes can be simplified. As concrete examples the decomposition of polynomials and truncated normal distribution are computed and graphically illustrated. Such interval measures lead to localized spherical probability measures, and as a convex expression of probability measure is a Jensen type inequality for spherical random variables. Based on the spherical information entropy models, one dependent on the domain partition and the other not, we propose three minimization formulations for the probabilistic finite element method, including one that mixes the least-squares and entropy models.

Greedy training algorithms for neural networks and applications to PDEs

Recently, neural networks have been widely applied for solving partial differential equations (PDEs). Although such methods have been proven remarkably successful on practical engineering problems, they have not been shown, theoretically or empirically, to converge to the underlying PDE solution with arbitrarily high accuracy. The primary difficulty lies in solving the highly non-convex optimization problems resulting from the neural network discretization, which are difficult to treat both theoretically and practically. It is our goal in this work to take a step toward remedying this. For this purpose, we develop a novel greedy training algorithm for shallow neural networks. Our method is applicable to both the variational formulation of the PDE and also to the residual minimization formulation pioneered by physics informed neural networks (PINNs). We analyze the method and obtain a priori error bounds when solving PDEs from the function class defined by shallow networks, which rigorously establishes the convergence of the method as the network size increases. Finally, we test the algorithm on several benchmark examples, including high dimensional PDEs, to confirm the theoretical convergence rate. Although the method is expensive relative to traditional approaches such as finite element methods, we view this work as a proof of concept for neural network-based methods, which shows that numerical methods based upon neural networks can be shown to rigorously converge.

Monolithic parallel overlapping Schwarz methods in fully-coupled nonlinear chemo-mechanics problems

We consider the swelling of hydrogels as an example of a chemo-mechanical problem with strong coupling between the mechanical balance relations and the mass diffusion. The problem is cast into a minimization formulation using a time-explicit approach for the dependency of the dissipation potential on the deformation and the swelling volume fraction to obtain symmetric matrices, which are typically better suited for iterative solvers. The MPI-parallel implementation uses the software libraries deal.II, p4est and FROSch (Fast of Robust Overlapping Schwarz). FROSch is part of the Trilinos library and is used in fully algebraic mode, i.e., the preconditioner is constructed from the monolithic system matrix without making explicit use of the problem structure. Strong and weak parallel scalability is studied using up to 512 cores, considering the standard GDSW (Generalized Dryja-Smith-Widlund) coarse space and the newer coarse space with reduced dimension. The FROSch solver is applicable to the coupled problems within in the range of processor cores considered here, although numerical scalablity cannot be expected (and is not observed) for the fully algebraic mode. In our strong scalability study, the average number of Krylov iterations per Newton iteration is higher by a factor of up to six compared to a linear elasticity problem. However, making mild use of the problem structure in the preconditioner, this number can be reduced to a factor of two and, importantly, also numerical scalability can then be achieved experimentally. Nevertheless, the fully algebraic mode is still preferable since a faster time to solution is achieved.

Determination of optimal potential parameters for the self-assembly of various lattice structures

We present a gradient-based optimization method that precisely designs simplified isotropic potentials for self-assembly of targeted lattice structures. An ansatz potential and its constraints are constructed to directly reflect the characteristics of a Bravais lattice for the target crystal, and the method is able to design the simplified potential systematically and smoothly. The potential is simplified with a Gaussian function in reciprocal space to minimize the information loss caused by Fourier transform in each design update. Design optimization formulation is derived using design sensitivity to relative entropy, employing a Fourier-Filtered Relative Entropy Minimization (FF-REM) formulation embedded in LAMMPS code to be used as an optimization tool. The potential obtained through this optimization method can successfully self-assemble low-coordinated crystals, such as square, triangle, honeycomb, and Kagome lattices, without further smoothing techniques. It turns out that completely removing the competitive alternative structures is an important condition for improving stability in self-assembling the target lattice in various density ranges.

Model reduction of convection-dominated partial differential equations via optimization-based implicit feature tracking

This work introduces a new approach to reduce the computational cost of solving partial differential equations (PDEs) with convection-dominated solutions: model reduction with implicit feature tracking. Traditional model reduction techniques use an affine subspace to reduce the dimensionality of the solution manifold and, as a result, yield limited reduction and require extensive training due to the slowly decaying Kolmogorov n-width of convection-dominated problems. The proposed approach circumvents the slowly decaying n-width limitation by using a nonlinear approximation manifold systematically defined by composing a low-dimensional affine space with a space of bijections of the underlying domain. Central to the implicit feature tracking approach is a residual minimization problem over the reduced nonlinear manifold that simultaneously determines the reduced coordinates in the affine space and the domain mapping that minimize the residual of the unreduced PDE discretization. This is analogous to standard minimum-residual reduced-order models, except instead of only minimizing the residual over the affine subspace of PDE states, our method enriches the optimization space to also include admissible domain mappings. The nonlinear trial manifold is constructed by using the proposed residual minimization formulation to determine domain mappings that cause parametrized features to align in a reference domain for a set of training parameters. Because the feature is stationary in the reference domain, i.e., the convective nature of solution removed, the snapshots are effectively compressed to define an affine subspace. The space of domain mappings, originally constructed using high-order finite elements, is also compressed in a way that ensures the boundaries of the original domain are maintained. Several numerical experiments are provided, including transonic and supersonic, inviscid, compressible flows, to demonstrate the potential of the method to yield accurate approximations to convection-dominated problems with limited training.

Stochastic absolute value equations

We propose a new kind of stochastic absolute value equations involving absolute values of variables. By utilizing an equivalence relation to stochastic bilinear program, we investigate the expected value formulation for the proposed stochastic absolute value equations. We also consider the expected residual minimization formulation for the proposed stochastic absolute value equations. Under mild assumptions, we give the existence conditions for the solution of the stochastic absolute value equations. The solution of the stochastic absolute value equations can be gotten by solving the discrete minimization problem. And we also propose a smoothing gradient method to solve the discrete minimization problem. Finally, the numerical results and some discussions are given.

Robust Optimal Classification Trees against Adversarial Examples

Decision trees are a popular choice of explainable model, but just like neural networks, they suffer from adversarial examples. Existing algorithms for fitting decision trees robust against adversarial examples are greedy heuristics and lack approximation guarantees. In this paper we propose ROCT, a collection of methods to train decision trees that are optimally robust against user-specified attack models. We show that the min-max optimization problem that arises in adversarial learning can be solved using a single minimization formulation for decision trees with 0-1 loss. We propose such formulations in Mixed-Integer Linear Programming and Maximum Satisfiability, which widely available solvers can optimize. We also present a method that determines the upper bound on adversarial accuracy for any model using bipartite matching. Our experimental results demonstrate that the existing heuristics achieve close to optimal scores while ROCT achieves state-of-the-art scores.

A spatio-temporal adaptive phase-field fracture method

We present an energy-preserving mechanic formulation for dynamic quasi-brittle fracture in an Eulerian–Lagrangian formulation, where a second-order phase-field equation controls the damage evolution. The numerical formulation adapts in space and time to bound the errors, solving the mesh-bias issues these models typically suffer. The time-step adaptivity estimates the temporal truncation error of the partial differential equation that governs the solid equilibrium. The second-order generalized-α time-marching scheme evolves the dynamic system. We estimate the temporal error by extrapolating a first-order approximation of the present time-step solution using previous ones with backward difference formulas; the estimate compares the extrapolation with the time-marching solution. We use an adaptive scheme built on a residual minimization formulation in space. We estimate the spatial error by enriching the discretization with elemental bubbles; then, we localize an error indicator norm to guide the mesh refinement as the fracture propagates. The combined space and time adaptivity allows us to use low-order linear elements in problems involving complex stress paths. We efficiently and robustly use low-order spatial discretizations while avoiding mesh bias in structured and unstructured meshes. We demonstrate the method’s efficiency with numerical experiments that feature dynamic crack branching, where the capacity of the adaptive space–time scheme is apparent. The adaptive method delivers accurate and reproducible crack paths on meshes with fewer elements.

Prediction models with graph kernel regularization for network data

Traditional regression methods typically consider only covariate information and assume that the observations are mutually independent samples. However, samples usually come from individuals connected by a network in many modern applications. We present a risk minimization formulation for learning from both covariates and network structure in the context of graph kernel regularization. The formulation involves a loss function with a penalty term. This penalty can be used not only to encourage similarity between linked nodes but also lead to improvement over traditional regression models. Furthermore, the penalty can be used with many loss-based predictive methods, such as linear regression with squared loss and logistic regression with log-likelihood loss. Simulations to evaluate the performance of this model in the cases of low dimensions and high dimensions show that our proposed approach outperforms all other benchmarks. We verify this for uniform graph, nonuniform graph, balanced-sample, and unbalanced-sample datasets. The approach was applied to predicting the response values on a ‘follow’ social network of Tencent Weibo users and on two citation networks (Cora and CiteSeer). Each instance verifies that the proposed method combining covariate information and link structure with the graph kernel regularization can improve predictive performance.

An active contour model using matched filter and Hessian matrix for retinal vessels segmentation

Medical image analysis, especially of the retina, plays an important role in diagnostic decision support tools. The properties of retinal blood vessels are used for disease diagnoses such as diabetes, glaucoma, and hypertension. There are some challenges in the utilization of retinal blood vessel patterns such as low contrast and intensity inhomogeneities. Thus, an automatic algorithm for vessel extraction is required. Active contour is a strong method for edge extraction. However, it cannot extract thin vessels and ridges very well. In this research, we propose an improved active contour method that uses discrete wavelet transform for energy minimization to solve this problem. The minimization formula terminates segmentation into two regions, foreground and background. We found out that foreground pixels are more important than background. Therefore, we change the minimization formulation in such a way that gives more weight to the foreground. The contour edge has orientation in each region. The wavelet terms in the minimization formula help to detect edge direction in horizontal, vertical, and diagonal orientations. Since bright and dark lesions do not have direction, these terms can decrease false vessel detection. The second part of the innovation is called the optimization process, which works instead of reinitialization. Sometimes, the evolution in the iterated process destroys the stability of the evolution. To prevent the destruction of the stability of evolution, we used an optimization process formula whose task is to keep contour on the edges of the image. The performance of the proposed algorithms is compared and analyzed on five databases. The values achieved are 94.3%, 73.36%, and 97.41% for accuracy, sensitivity, and specificity, respectively, on the DRIVE dataset, and the proposed algorithm is comparable to the state-of-the-art approaches.

Multimodal correlations-based data clustering

<p style='text-indent:20px;'>This work proposes a novel technique for clustering multimodal data according to their information content. Statistical correlations present in data that contain similar information are exploited to perform the clustering task. Specifically, multiset canonical correlation analysis is equipped with norm-one regularization mechanisms to identify clusters within different types of data that share the same information content. A pertinent minimization formulation is put forth, while block coordinate descent is employed to derive a batch clustering algorithm which achieves better clustering performance than existing alternatives. Relying on subgradient descent, an online clustering approach is derived which substantially lowers computational complexity compared to the batch approach, while not compromising significantly the clustering performance. It is established that for an increasing number of data the novel regularized multiset framework is able to correctly cluster the multimodal data entries. Further, it is proved that the online clustering scheme converges with probability one to a stationary point of the ensemble regularized multiset correlations cost having the potential to recover the correct clusters. Extensive numerical tests demonstrate that the novel clustering scheme outperforms existing alternatives, while the online scheme achieves substantial computational savings.</p>

Iterative regularization for constrained minimization formulations of nonlinear inverse problems

In this paper we study the formulation of inverse problems as constrained minimization problems and their iterative solution by gradient or Newton type methods. We carry out a convergence analysis in the sense of regularization methods and discuss applicability to the problem of identifying the spatially varying diffusivity in an elliptic PDE from different sets of observations. Among these is a novel hybrid imaging technology known as impedance acoustic tomography, for which we provide numerical experiments.