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A localized Fourier collocation method is proposed for solving certain types of elliptic boundary value problems. The method first discretizes the entire domain into a set of overlapping small subdomains, and then in each of the subdomains, the unknown functions and their derivatives are approximated using the pseudo‐spectral Fourier collocation method. The key idea of the present method is to combine the merits of the quick convergence of the pseudo‐spectral method and the high sparsity of the localized discretization technique to yield a new framework that may be suitable for large‐scale simulations. The present method can be viewed as a competitive alternative for solving numerically large‐scale boundary value problems with complex‐shape geometries. Preliminary numerical experiments involving Poisson, Helmholtz, and modified‐Helmholtz equations in both two and three dimensions are presented to demonstrate the accuracy and efficiency of the proposed method.

This work is focused on the topology optimization related to harmonic responses for large-scale problems. A comparative study is made among mode displacement method (MDM), mode acceleration method (MAM) and full method (FM) to highlight their effectiveness. It is found that the MDM results in the unsatisfactory convergence due to the low accuracy of harmonic responses, while MAM and FM have a good accuracy and evidently favor the optimization convergence. Especially, the FM is of superiority in both accuracy and efficiency under the excitation at one specific frequency; MAM is preferable due to its balance between the computing efficiency and accuracy when multiple excitation frequencies are taken into account.

In this paper, based on the hyperplane projection technique, we propose a three-term conjugate gradient method for solving nonlinear monotone equations with convex constraints. Due to the derivative-free feature and lower storage requirement, the proposed method can be applied to the solution of large-scale non-smooth nonlinear monotone equations. Under some mild assumptions, the global convergence is proved when the line search fulfils the backtracking line search condition. Moreover, we prove that the proposed method is R-linearly convergent. Numerical results show that our method is competitive and efficient for solving large-scale nonlinear monotone equations with convex constraints.

We propose a new density-based topology optimization method for applications where fluid–structure interaction (FSI) plays a significant role. The method utilizes the jump in density between neighboring finite elements to implicitly track the FSI boundary and the FSI load. A pressure penalty term is introduced into the Navier–Stokes equations to mitigate the formation of internal pressurized holes, providing a more accurate representation of the physics and enhancing manufacturability. The method is implemented using a parallelized computational framework that enables efficient optimization of large-scale 3D problems. High-resolution discretization, combined with filtering techniques, minimizes intermediate densities and achieves detailed, binary structures that accurately model the FSI load. This is then exemplified using a classic FSI benchmark (the wall problem) with different objective functions and constraints. A relevant engineering example is then shown, maximizing fluid performance with a mechanical constraint. The approach demonstrates good convergence and provides conceptually robust designs with potential for further refinement.

The bat algorithm (BA) is one of prominent swarm-based algorithm that has the suitability in solving only small dimension engineering problems and suffers from drawback of getting trapped in local minimum with slow convergence for multi-dimensional problems. In the context of improving its applicability in solving large scale and constrained engineering design problems, this paper presents a novel upgraded bat algorithm with cuckoo search and Sugeno inertia weight (UBCSIW). In the proposed UBCSIW algorithm, first, the bat algorithm with its competence to exploit the optimal solutions in search space is combined with cuckoo search with its ability to explore best solution globally using Levy flight in the search space. Secondly, a new velocity and position search equation is incorporated in which the bat searches around the best candidate solution. This step helps in establishing adequate balance between exploration and exploitation capability and improving the performance effectively by employing greedy selection to choose the best candidate solution. Finally, Sugeno fuzzy inertia weight is introduced in the velocity updation equation, boosting the flexibility and diversity of bat population and results in stability of results. The effectiveness of the proposed UBCSIW algorithm is tested on 16 standard benchmark functions (unimodal and multimodal) with different dimensions, 12 CEC2015 test functions and 7 well-known constrained engineering design problems. The outputs of the proposed UBCSIW algorithm are validated by comparison with classical BA and other swarm-based state-of-the art algorithms. The simulation results show that proposed UBCSIW algorithm achieves highly competitive results in terms of higher optimization accuracy and improved convergence that outperforms basic BA in all twenty-eight test functions while performs better than other competitive algorithms in 24 functions (13 benchmark and 11 CEC2015 functions).

This paper presents a computationally efficient multi-resolution topology optimization framework by establishing a novel bi-directional evolutionary structure optimization (BESO) method based on extended finite element method (XFEM). In the proposed framework, the high-resolution designs preserving the topological complexity can be obtained with low degree of freedoms (DOFs). The implementation of the presented multi-resolution optimization framework takes good use of the ability of XFEM at accurately modeling material discontinuities within one element. On the basis of XFEM, a strategy of triangulated partition and a new material interpolation model are introduced to represent the finer material distribution. We employ the coarser finite element (FE) mesh to perform the finite element analysis, the sub-parts partitioned from finite elements to describe material properties and the nodal design variables to perform the optimization. To circumvent artificially stiff patterns, a modified sensitivity filter is applied to regularize the solution. The effectiveness and high efficiency of the presented approach are highlighted by typical 2D and 3D examples.

Often, parameter estimation problems of parameter-dependent PDEs involve multiple right-hand sides. The computational cost and memory requirements of such problems increase linearly with the number of right-hand sides. For many applications this is the main bottleneck of the computation. In this paper we show that problems with multiple right-hand sides can be reformulated as stochastic programming problems by combining the right-hand sides into a few \"simultaneous\" sources. This effectively reduces the cost of the forward problem and results in problems that are much cheaper to solve. We discuss two solution methodologies: namely sample average approximation and stochastic approximation. To illustrate the effectiveness of our approach we present two model problems, direct current resistivity and seismic tomography. [PUBLICATION ABSTRACT]

In the complex optimization scenarios of real life, we often need to not only weigh the conflicts between multi-objectives but also face the challenges brought by large-scale decision variables. When the scale of decision variables increases sharply, the search space will expand exponentially, which makes it difficult for the algorithm to traverse the whole search space under limited resources, so that the number of local optimal solutions increases sharply. The so-called ‘dimension disaster’ appears. To effectively deal with this problem, this paper proposes a large-scale multi-objective brainstorm optimization algorithm (LMaOBSO) based on low complexity decision variable classification method and improved penalty boundary intersection strategy. The algorithm uses a low-complexity tree sorting variable classification method to divide the decision variables into converges and diversity variables quickly. At the same time, the convergent variables are optimized by the improved penalty boundary crossover strategy to solve the problems existing in large-scale optimization problems.