This chapter introduces the finite element method (FEM) as a tool for solution of classical electromagnetic problems. Although we discuss the main points in the application of the finite element method to electromagnetic design, including formulation and implementation, those who seek deeper understanding of the finite element method should consult some of the works listed in the bibliography section.

Introduction to the Finite Element Method

Computational Fluid Dynamics: Principles and Applications, Third Edition presents students, engineers, and scientists with all they need to gain a solid understanding of the numerical methods and principles underlying modern computation techniques in fluid dynamics By providing complete coverage of the essential knowledge required in order to write codes or understand commercial codes, the book gives the reader an overview of fundamentals and solution strategies in the early chapters before moving on to cover the details of different solution techniques

	This updated edition includes new worked programming examples, expanded coverage and recent literature regarding incompressible flows, the Discontinuous Galerkin Method, the Lattice Boltzmann Method, higher-order spatial schemes, implicit Runge-Kutta methods and parallelization

	An accompanying companion website contains the sources of 1-D and 2-D Euler and Navier-Stokes flow solvers (structured and unstructured) and grid generators, along with tools for Von Neumann stability analysis of 1-D model equations and examples of various parallelization techniques



	
		Will provide you with the knowledge required to develop and understand modern flow simulation codes
	
		Features new worked programming examples and expanded coverage of incompressible flows, implicit Runge-Kutta methods and code parallelization, among other topics
	
		Includes accompanying companion website that contains the sources of 1-D and 2-D flow solvers as well as grid generators and examples of parallelization techniques

Computational Fluid Dynamics: Principles and Applications Ed. 3

A multivariate interpolation scheme, using radial basis functions, is presented, which results in a completely unified formulation for the fluid–structure interpolation and mesh motion problems. The method has several significant advantages. Primarily, all volume mesh, structural mesh, and flow-solver type dependence is removed, and all operations are performed on totally arbitrary point clouds of any form. Hence, all connectivity and user-input requirements are removed from the computational fluid dynamics–computational structural dynamics (CFD–CSD) coupling problem, as only point clouds are required to determine the coupling. Also, it may equally well be applied to structured and unstructured grids, or structural and aerodynamic grids that intersect, again because no connectivity information is required. Furthermore, no expensive computations are required during an unsteady simulation, just matrix–vector multiplications, since the required dependence relations are computed only once prior to any simulation and then remain constant. This property means that the method is both perfectly parallel, since only the data relevant to each structured block or unstructured partition are required to move those points, and totally independent from the flow solver. Hence, a completely generic ‘black box’ tool can be developed, which is ideal for use in an optimization approach. Aeroelastic behaviour of the Brite–Euram MDO wing is analysed in terms of both static deflection and dynamic responses, and it is demonstrated that responses are strongly dependent on the exact CFD–CSD interpolation used. Mesh quality is also examined during the motion resulting from a large surface deformation. Global grid quality is shown to be preserved well, with local grid orthogonality also being maintained well, particularly at and near the moving surface, where the original orthogonality is retained. Copyright © 2007 John Wiley & Sons, Ltd.

Unified fluid-structure interpolation and mesh motion using radial basis functions

The efficiency of building a surrogate model for the output of a computer code can be dramatically improved via variable-fidelity surrogate modeling techniques. In this article, a hierarchical kriging model is proposed and used for variable-fidelity surrogate modeling problems. Here, hierarchical kriging refers to a surrogate model of a highfidelity function that uses a kriging model of a sampled lower-fidelity function as a model trend. As a consequence, the variation in the lower-fidelity data is mapped to the high-fidelity data, and a more accurate surrogate model for the high-fidelity function is obtained. A self-contained derivation of the hierarchical kriging model is presented. The proposed method is demonstrated with an analytical example and used for modeling the aerodynamic data of an RAE 2822 airfoil and an industrial transport aircraft configuration. The numerical examples show that it is efficient, accurate, and robust. It is also observed that hierarchical kriging provides a more reasonable mean-squared-error estimation than traditional cokriging. It can be applied to the efficient aerodynamic analysis and shape optimization of aircraft or any other research areas where computer codes of varying fidelity are in use.

Hierarchical Kriging Model for Variable-Fidelity Surrogate Modeling

Metamodeling is becoming a rather popular means to approximate the expensive simulations in today’s complex engineering design problems since accurate metamodels can bring in a lot of benefits. The metamodel accuracy, however, heavily depends on the locations of the observed points. Adaptive sampling, as its name suggests, places more points in regions of interest by learning the information from previous data and metamodels. Consequently, compared to traditional space-filling sampling approaches, adaptive sampling has great potential to build more accurate metamodels with fewer points (simulations), thereby gaining increasing attention and interest by both practitioners and academicians in various fields. Noticing that there is a lack of reviews on adaptive sampling for global metamodeling in the literature, which is needed, this article categorizes, reviews, and analyzes the state-of-the-art single−/multi-response adaptive sampling approaches for global metamodeling in support of simulation-based engineering design. In addition, we also review and discuss some important issues that affect the success of an adaptive sampling approach as well as providing brief remarks on adaptive sampling for other purposes. Last, challenges and future research directions are provided and discussed.

A survey of adaptive sampling for global metamodeling in support of simulation-based complex engineering design

Elements of computational fluid dynamics on block structured grids using implicit solvers

Non-linear aeroelastic prediction for aircraft applications

Transonic aeroelastic simulation for instability searches and uncertainty analysis

The generation of a tabular aerodynamic model for design related flight dynamics studies, based on simulation generated data, is considered. The framework described accommodates two design scenarios. The first emphasizes the representation of the aerodynamic nonlinearities, and is based on sampling. The second scenario assumes incremental change from an initial geometry, for which a hi-fidelity model from the first scenario is available. In this case data fusion is used to update the model. In both cases, Kriging is used to interpolate the samples computed using simulation. A commercial jet test case, using DATCOM as a source of data, is computed to illustrate the sampling and fusion. Future application using Computational Fluid Dynamics as the source of data is considered.

/pdf/accelerating-the-numerical-generation-of-aerodynamic-models-2kpnb4q0df.pdf

Accelerating the Numerical Generation of Aerodynamic Models for Flight Simulation

The computation of the aerodynamic forces arising from forced periodic motions is required for the generation of dynamic terms in models for flight simulation. The periodicity can be used to avoid using fully unsteady calculations by using the harmonic balance method. The current paper develops an implicit solver for the harmonic balance equations. The method is tested on two transonic test cases and evaluation is made against the unsteady simulation results. The first caseis for the pitching NACA 0012aerofoil. The second is for forced pitching of the F-5 wing with a wing tip launcher and missile. A reduction in computational time by one order of magnitude compared with the unsteady solver is obtained. Nomenclature A = matrix in frequency domain equation c = chord D = matrix in harmonic balance equation E = transformation matrix between frequency and time domains e = energy F, G, H = convective fluxes I = residual of semidiscrete system I = identity matrix k = reduced frequency nH = number of harmonics p = pressure R = residual vector T = period t = time u, v, w = Cartesian velocity components W = conserved variables � = angle of attack � t = pseudo time step

Mark Woodgate

Papers

Elements of computational fluid dynamics on block structured grids using implicit solvers

Non-linear aeroelastic prediction for aircraft applications

Transonic aeroelastic simulation for instability searches and uncertainty analysis

Accelerating the Numerical Generation of Aerodynamic Models for Flight Simulation

Implicit Harmonic Balance Solver for Transonic Flow with Forced Motions