Mathematical studies in rigorous grating theory

A model-based method has been developed for the performance simulation and conceptual design of rocket-type pulse detonation engines (PDEs). A reduced-order model (ROM) has been generated based on the high order singular value decomposition of a data tensor obtained from CFD computations. This ROM could be used to solve the direct (performance) and inverse (design) problems in the context of the early phases of pulse detonation engine design. Output performance parameters are predicted from prescribed input operation/geometry parameters in the direct problem, and vice versa in the inverse problem. The focus of this method is industrial application in situations where large parametric searches are to be performed with a reasonable level of fidelity at a low computational cost. It was found that the performance and conceptual design tool thus developed provides results that deviate, on average, by less than 10% from the CFD results. Regarding practical implementation, the method allows for shifting the heavier computational load off-line. In this way, when working on-line, the user can obtain results in less than a second for every single case. The main contribution of this study is showing that a model-based approach that combines CFD and tensor decomposition has the potential to extract a maximum of information from a given computational effort. This characteristic makes the method of interest for early design phases in the aerospace industry.

In this work, an inverse design method for multiinput multi-output (MIMO) metastructured devices is developed. Large-scale inverse design problems are difficult to solve directly and often require heuristic methods or design optimization to find a solution. Inherent errors introduced by heuristic methods makes design optimization a more promising route to the realization of high performance devices. Here, a fast frequency domain solver for grids of Y-parameter matrices is developed. The solver is used together with an adjoint-based optimization routine to solve inverse metastructured design problems. The design procedure is demonstrated through the realization of a planar beamforming network for a multi-beam antenna.

Purpose – The purpose of this paper is to present a simple approach for calculation of the sensitivities in the free-form inverse design problems. The approach is based on the analogy with the similar tasks used in the signal-processing analysis. In the proposed case it is not required to solve an adjoint problem as in the most of the similar optimization tasks. The simulation engine used in the background is a Fast Boundary Element Method. The approach is validated on some known benchmark problems. Design/methodology/approach – Inverse design is recognized nowadays as a crucial scientific grand challenge. Contrary to the conventional approach (“Given the structure, find the properties”) it purses a new paradigm (“Given the desired property, find the structure”). Inverse class of problems has a broad application area, from the material-, medical-, bio- to the engineering-class of problems. When dealing with the inverse design in free-form optimization of the engineering problems the typical approach is to calculate the adjoint problem. Calculation of the adjoint problem mostly requires the costly calculation of the gradients, which makes the whole optimization procedure rather expensive due to the high computational burden required for their solution. Findings – In this paper it is proposed a novel Simple Sensitivity Approach to get in a fast way the response (sensitivity) function of the analyzed structure. The simulation engine used in the background is the Fast Boundary Element Method. Originality/value – Novel approach for inverse design when performing the free-form optimization of engineering problems.

Background. The purpose of this study is to develop an effective non-invasive method for determining the properties of the object under consideration. Material and methods. For these purposes, the inverse diffraction problem is solved using combined or generalized computational grids. The paper presents a description of the direct and inverse problems. Results and conclusions. The result of solving the direct problem is obtained as a solution to the corresponding volume integral equation. To solve the inverse problem, a two-step method is used. A detailed description of the numerical method is presented. Numerical results are obtained and solutions to the problem with noisy and non-noisy input data are compared.

This paper presents a methodology for the solution of an inverse solidification design problem in the presence of natural convection. In particular, the boundary heat flux q0 in the fixed mold wall, δΩ0, is calculated such that a desired freezing front velocity and shape are obtained. As the front velocity together with the flux history qms on the solid side of the freezing front play a determinant role in the obtained cast structure, the potential applications of the proposed methods to the control of casting processes are enormous.The proposed technique consists of first solving a direct natural convection problem of the liquid phase in an a priori known shrinking cavity, ΩL(t), before solving an ill‐posed inverse design conduction problem in the solid phase in an a priori known growing region, ΩS(t). The direct convection problem is used to evaluate the flux qml in the liquid side of the freezing front. A front tracking deforming finite element technique is employed. The flux qml can be used together with the Stefan condition to provide the freezing interface flux qms in the solid side of the front. As such, two boundary conditions (flux qms and freezing temperature θm) are especified along the (known) freezing interface δΩI(t). The developed design technique uses the adjoint method to calculate in L2 the derivative of the cost functional, ∥θm – θ(x, t; q0)∥, that expresses the square error between the calculated temperature θ(x, t; q0) in the solid phase along δΩI(t) and the given melting temperature. The minimization of this cost functional is performed by the conjugate gradient method via the solutions of the direct, sensitivity and adjoint problems. A front tracking finite element technique is employed in this inverse analysis. Finally, an example is presented for the solidification of a superheated incompressible liquid aluminium, where the effects of natural convection in the moving interface shape are controlled with a proper adjustment of the cooling boundary conditions.

A class of optimal shape design problems is studied in this paper. The boundary shape of a domain is determined such that the solution of the underlying partial differential equation matches, as well as possible, a given desired state. In the original optimal shape design problem, the variable domain is parameterized by a class of functions in such a way that the optimal design problem is changed to an optimal control problem on a fixed domain. Then, the resulting distributed control problem is embedded in a measure theoretical form, in fact, an infinite-dimensional linear programming problem. The optimal measure representing the optimal shape is approximated by a solution of a finite-dimensional linear programming problem. The method is evaluated via a numerical example.

In this paper, we develop a non‐iterative way to solve biharmonic boundary value problems by using a radial basis meshless method. This is an original application of meshless method to solving inverse problems without any iteration, since traditional numerical methods for inverse boundary value problems mainly are iterative and hence very time‐consuming. Numerical examples are presented for inverse biharmonic boundary value problems and corresponding direct problems, since solving direct problems is a preliminary step for inverse problems. All our examples of direct and inverse problems are solved within seconds in CPU time on a standard PC, which makes our proposed technique a great potential candidate for wide‐spread applications to other inverse problems. Copyright © 2004 John Wiley & Sons, Ltd.

We report a series of deep learning models to solve complex forward and inverse design problems in molecular modeling and design. Using both diffusion models inspired by nonequilibrium thermodynamics and attention-based transformer architectures, we demonstrate a flexible framework to capture complex chemical structures. First trained on the Quantum Machines 9 (QM9) dataset and a series of quantum mechanical properties (e.g., homo, lumo, free energy, and heat capacity), we then generalize the model to study and design key properties of deep eutectic solvents (DESs). In addition to separate forward and inverse models, we also report an integrated fully prompt-based multi-task generative pretrained transformer model that solves multiple forward, inverse design, and prediction tasks, flexibly and within one model. We show that the multi-task generative model has the overall best performance and allows for flexible integration of multiple objectives, within one model, and for distinct chemistries, suggesting that synergies emerge during training of this large language model. Trained jointly in tasks related to the QM9 dataset and DESs, the model can predict various quantum mechanical properties and critical properties to achieve deep eutectic solvent behavior. Several combinations of DESs are proposed based on this framework.

Numerical method for shape optimization using BEM

Deep learning surrogate models have shown promise in modeling complex physical phenomena such as photonics, fluid flows, molecular dynamics and material properties. However, standard neural networks assume finite-dimensional inputs and outputs, and hence, cannot withstand a change in resolution or discretization between training and testing. We introduce Fourier neural operators that can learn operators, which are mappings between infinite dimensional spaces. They are discretization-invariant and can generalize beyond the discretization or resolution of training data. They can efficiently solve partial differential equations (PDEs) on general geometries. We consider a variety of PDEs for both forward modeling and inverse design problems, as well as show practical gains in the lithography domain.

This work is devoted to the development and implementation of a two-step method for solving the vector three-dimensional inverse diffraction problem on an inhomogeneous dielectric scatterer having the form of a hemisphere characterized by piecewise constant permittivity. The original boundary value problem for Maxwell’s equations is reduced to a system of integro-differential equations. An integral formulation of the vector inverse diffraction problem is proposed and the uniqueness of the solution of the first-kind integro-differential equation in special function classes is established. A two-step method for solving the vector inverse diffraction problem on the hemisphere is developed. Unlike traditional approaches, the two-step method for solving the inverse problem is non-iterative and does not require knowledge of the exact initial approximation. Consequently, there are no issues related to the convergence of the numerical method. The results of calculations of approximate solutions to the inverse problem are presented. It is shown that the two-step method is an efficient approach to solving vector problems in near-field tomography.

A continuous dynamic model for the development of a gas field is studied. Two mathematical problems are posed, solved and analyzed. In the direct problem, we are looking for the maximum accumulated profit on a set of temporary functions of a certain type. In the inverse optimization problem, we define the entire set of planning horizons. We subject the assigned tasks to a comprehensive analysis.

Exact solution for an optimal impermeable parachute problem

Interpretable Forward and Inverse Design of Particle Spectral Emissivity Using Common Machine-Learning Models

Nonlinear optimal regulators are discussed, and some useful properties are isolated. An inverse problem of nonlinear regulator design is posed and solved.