# Syllabus For Mech568 Computational Methods For Mechanical

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SYLLABUS FOR MECH568 COMPUTATIONAL METHODS FOR MECHANICAL ENGINEERING
Instructor: Xinfeng Gao
1. Course Description
In ME, many research areas have an associated computational aspect. Computational ﬂuid dynamic simulations (gas dynamics, ﬂuid mechanics, bio-ﬂuids, etc.), ﬁnite element simulations, and particle interaction simulations are applied regularly to study ﬂuid-thermal, ﬂuid-structure, and mechanical systems at all scales. In addition, new computational methods and techniques are being developed to advance the state-of-the art in simulation capacity.
The goal of the software user is to generate a solution that is useful, trustworthy, and accurate; the goal of the software developer is to make this as likely as possible. A great deal of knowledge and expertise of the fundamentals in computational methods is needed not only to develop algorithms and models, but also to apply them successfully.
This course is designed as a computational core course for ME graduate students who seek a deep understanding of the fundamental principles which provide the foundation for the software and algorithms used in ME. The purpose of this course is to provide the foundation needed to achieve the above goals of both users and developers.
2. Course Objectives
Upon completion of this course, the students will understand and systematize numerical solution techniques for the partial diﬀerential equations governing the physics of mechanical engineering problems. They will learn the mathematical structure which could be used to describe the behavior and results of most numerical methods in common used in these problems. Students will write codes using Matlab, C++ and FORTRAN.
3. Course Materials
Topics 1 Introduction 2 Conservation Laws and Model Equations
Conservation Laws, Euler Equations, Navier-Stokes Equations Linear Convection and Diﬀusion Equation Linear Hyperbolic Systems Diﬀerential Form and Solution in Wave Space 3 Finite-Diﬀerence Approximations Space Derivative Approximations Finite-Diﬀerence Operators Constructing Diﬀerencing Schemes of Any Order Fourier Error Analysis Diﬀerence Operators at Boundaries 4 The Semi-Discrete Approach Reduction of PDE’s to ODE’s Real Space and Eigenspace 5 Finite-Volume Methods Model Equations in Integral Form Multidimensional Examples
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Topics 6 Finite-Element Methods
Approximation of Elliptic Problems Piecewise Polynomial Approximation A Posterior Error Analysis Evolution Problems 7 Time-Marching Methods for ODE’s Converting Time-Marching Methods to O∆E’s The λ − σ Relation Accuracy Measures of Time-Marching Methods Linear Multistep Methods, Predictor-Corrector Methods Implementation of Implicit Methods 8 Stability of Linear Systems Dependence on the Eigensystem Inherent Stability of ODE’s Numerical Stability of O∆E’s Time-Space Stability and Convergence of O∆E’s Numerical Stability Concepts in the Complex σ−Plane Numerical Stability Concepts in the Complex λhPlane Fourier Stability Analysis Consistency 9 Choosing a Time-Marching Method Stiﬀness Deﬁnition for ODE’s and Relation of Stiﬀness to Space Mesh Size Practical Considerations for Comparing Methods Comparing the Eﬃciency of Explicit Methods Coping with Stiﬀness 10 Relaxation Methods Classical Relaxation The ODE Approach to Classical Relaxation Eigensystems of the Classical Methods Nonstationary Processes 11 Multigrid Eigenvector and Eigenvalue Identiﬁcation with Space Frequencies and Properties of the Iterative Methods A Two-Grid Process 12 Numerical Dissipation and Dispersion Upwind Schemes Artiﬁcial Dissipation
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