Résultats de recherche pour « High-fidelity numerical methods »

Lack of conservation has been the biggest drawback in meshfree generalized finite difference methods (GFDMs). In this paper, we present a novel modification of classical meshfree GFDMs to include local balances which produce an approximate conservation of numerical fluxes. This numerical flux conservation is done within the usual moving least squares framework. Unlike Finite Volume Methods, it is based on locally defined control cells, rather than a globally defined mesh. We present the application of this method to an advection diffusion equation and the incompressible Navier - Stokes equations. Our simulations show that the introduction of flux conservation significantly reduces the errors in conservation in meshfree GFDMs.

We couple the L1 discretization for Caputo derivative in time with spectral Galerkin method in space to devise a scheme that solves quasilinear subdiffusion equations. Both the diffusivity and the source are allowed to be nonlinear functions of the solution. We prove method's stability and convergence with spectral accuracy in space. The temporal order depends on solution's regularity in time. Further, we support our results with numerical simulations that utilize parallelism for spatial discretization. Moreover, as a side result we find asymptotic exact values of error constants along with their remainders for discretizations of Caputo derivative and fractional integrals. These constants are the smallest possible which improves the previously established results from the literature.

Gas turbine engines represent one of the most complex and sophisticated engineering systems in modern technology, with applications spanning power generation, aviation, and marine propulsion. The aerodynamic performance of these systems directly impacts their efficiency, reliability, and environmental footprint. This comprehensive review examines the revolutionary advancements in numerical methods that have transformed our understanding and analysis of gas turbine aerodynamics. The paper explores the evolution from traditional computational approaches to cutting-edge high-fidelity simulation techniques, including Direct Numerical Simulation (DNS), Large Eddy Simulation (LES), and hybrid RANS-LES methods. We provide an in-depth analysis of advanced turbulence modeling approaches, highlighting their theoretical foundations, implementation strategies, and practical applications in gas turbine environments. The review further investigates state-of-the-art aerothermodynamic analyses, focusing on conjugate heat transfer, film cooling optimization, and multi-physics coupling. Through detailed case studies and critical evaluation of current methodologies, this paper demonstrates how revolutionary numerical methods have redefined our understanding of flow physics, heat transfer mechanisms, and overall performance prediction in gas turbine systems. Finally, we discuss emerging technologies and future research directions, including exascale computing, digital twin implementation, and machine learning integration, that promise to further advance the field of gas turbine aerodynamics.

Left-invariant PDE-evolutions on the roto-translation group $SE(2)$ (and their resolvent equations) have been widely studied in the fields of cortical modeling and image analysis. They include hypo-elliptic diffusion (for contour enhancement) proposed by Citti & Sarti, and Petitot, and they include the direction process (for contour completion) proposed by Mumford. This paper presents a thorough study and comparison of the many numerical approaches, which, remarkably, is missing in the literature. Existing numerical approaches can be classified into 3 categories: Finite difference methods, Fourier based methods (equivalent to $SE(2)$-Fourier methods), and stochastic methods (Monte Carlo simulations). There are also 3 types of exact solutions to the PDE-evolutions that were derived explicitly (in the spatial Fourier domain) in previous works by Duits and van Almsick in 2005. Here we provide an overview of these 3 types of exact solutions and explain how they relate to each of the 3 numerical approaches. We compute relative errors of all numerical approaches to the exact solutions, and the Fourier based methods show us the best performance with smallest relative errors. We also provide an improvement of Mathematica algorithms for evaluating Mathieu-functions, crucial in implementations of the exact solutions. Furthermore, we include an asymptotical analysis of the singularities within the kernels and we propose a probabilistic extension of underlying stochastic processes that overcomes the singular behavior in the origin of time-integrated kernels. Finally, we show retinal imaging applications of combining left-invariant PDE-evolutions with invertible orientation scores.

Abstract In this paper a high‐fidelity aerodynamic model is presented for use in parametric studies of weapon aerodynamics. The method employs a reduced‐order model obtained from the proper orthogonal decomposition (POD) of an ensemble of computational fluid dynamics (CFD) solutions with varying parameters. This decomposition produces an optimal linear set of orthogonal basis functions that best describe the ensemble of numerical solutions. These solutions are then projected onto this set of basis functions to provide a finite set of scalar coefficients that represent the solutions. A pseudo‐continuous representation of these projection coefficients is constructed, which allows predictions to be made of parameter combinations not in the original set of observations. The paper explores the performance of a few design‐of‐experiment approaches for the generation of the initial ensemble of computational experiments. Response surface construction methods based on parametric and non‐parametric models for the pseudo‐continuous representation of the projection coefficients are also evaluated. The model has been applied to two‐flow problems related to high‐speed weapon aerodynamics, inviscid flow around a flare‐stabilized hypersonic projectile and supersonic turbulent flow around a fin‐stabilized projectile with drooping nose control. Comparisons of model predictions with high‐fidelity CFD simulations suggest that the POD provides a reliable and robust approach to the construction of reduced‐order models. The practicality of the model is shown to be sensitive to the technique used to generate the ensemble of observations from which the model is constructed, while the accuracy of the approach depends on the pseudo‐continuous representation of the projection coefficients. Copyright © 2009 John Wiley & Sons, Ltd.

The aim of this work is to study the numerical solution of the nonlinear Schrodinger problem using a combination between Witt basis and finite difference approximations. We construct a discrete fundamental solution for the non-stationary Schrodinger operator and we show the convergence of the numerical scheme. Numerical examples are given at the end of the paper.

Abstract Aerodynamic shape optimization technology is presented, using an efficient domain element parameterization approach. This provides a method that allows geometries to be parameterized at various levels, ranging from gross three‐dimensional planform alterations to detailed local surface changes. Design parameters control the domain element point locations and, through efficient global interpolation functions, deform both the surface geometry and corresponding computational fluid dynamics volume mesh, in a fast, high quality, and robust fashion. This results in total independence from the mesh type (structured or unstructured), and optimization independence from the flow‐solver is achieved by obtaining gradient information for an advanced gradient‐based optimizer by finite‐differences. Hence, the optimization tool can be used in conjunction with any flow‐solver and/or mesh generator. Results have been presented recently for two‐dimensional aerofoil cases, and shown impressive results; drag reductions of up to 45% were demonstrated using only 22 active design parameters. This paper presents the extension of these methods to three dimensions, with results for highly constrained optimization of a modern aircraft wing in transonic cruise. The optimization uses combined global and local parameters, giving 388 design variables, and produces a shock‐free geometry with an 18% reduction in drag, with the added advantage of significantly reduced root moments. Copyright © 2009 John Wiley & Sons, Ltd.

Abstract An adjoint‐based procedure to determine weaknesses, or, more generally, the material properties of structures is developed and tested. Given a series of load cases and corresponding displacement/strain measurements, the material properties are obtained by minimizing the weighted differences between the measured and computed values. The present paper proposes and tests techniques to minimize the number of load cases and sensors. Several examples show the viability, accuracy and efficiency of the proposed methodology and its potential use for high fidelity digital twins.