EGM6422 Advanced Analysis of Mechanical Systems

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Department of Mechanical and Materials Engineering

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This is the EGM 6422 Advanced Computational Engineering Analysis Spring 2018

Here is the (12/26/2017) updated syllabus for the course.

My office is in EC3442, and email address is levyez@fiu.edu

My tel. no. is 305-348-3643. My fax no. is the department fax no. 305-348-1932

Office Hours: Monday from 10am-12pm and Wednesday from 300-430pm

TA information: will be announced. If none, please see me in my office EC3442

The following reviews will be discussed initially

1. Review of Matrix Equations and their solutions: example of where used‑‑

Modelling of dynamic systems or curve fitting. Generation of systems of equations from the discretization process. Defining matrix equations.

a. Review of matrix notation, matrix norms, conditioning of matrices. What does the ill‑conditioning of matrices mean from an engineering point of view.

This material and all the linked materials provided, except where stated specifically, are copyrighted © Cesar Levy 2018 and is provided to the students of this course only. Use by any other individual without written consent of the author is forbidden.

b. Solution of Matrix equations using Gauss elimination, Gauss‑ Jordan and other methods (LU) decomposition.

c. Iterative techniques, convergence of iterative techniques, Jacobi and Gauss-Seidel iterative methods

d. Iterative techniques and relation to fixed point iteration; definition of the pseudoinverse; show the pseudoinverse for Jacobi and G-S method

Here is the video of the first lecture that is related to the first part of this review.

Here is the video of the second lecture that is related to this review.

Here is the video of the third lecture that is related to this review.

Here is the video of the fourth lecture that is related to this review and also the beginning of finite difference solution of differential equations

2. Implicit versus explicit methods for ODEs

Here is the video of the fifth lecture that is related to how to find the difference formulas for derivatives given the error we wish to achieve, how these formulas in governing equations give rise to the implicit method of solution, how the solution to the implicit method relates to the previous lectures, and how higher order differential equations can be solved by several first order differential equations that can be solved by the shooting method that incorporate the methods discussed in previous lectures.

3. Nonlinear Algebraic Sets of Equations-used to solve nonlinear PDEs

4. Ordinary Differential Equations

Single step methods available

Multistep methods available

Here is the video of the sixth lecture that is related to single step methods for solution of ODE’s both linear and nonlinear (Euler, Modified Euler, Runge Kutta, Runge Kutta Fehlberg) leading to Multistep Methods such as (Adams, Adams- Moulton, Milne’) and the solution of difference equations in comparison to real solution to the ODE’s, where applicable.

Here is the video of the seventh lecture that is related to multistep methods for solution of ODE’s both linear and nonlinear (such as Adams Method, Adams-Moulton, Milne’s methods) and a discussion of proving how the solutions of the difference equations generated by these numerical methods converge to the solutions of the differential equations. Also a discussion is given as to why Milne’s method converges some of the times and how the choice of the difference equation representation of the derivative can lead to conditional or unconditional stability of the numerical method.

Topics to be covered in the next lecture

5. Introduction to PDE's

What PDE's characterize

Classification of PDE's

Finite Difference Notation

6. Parabolic Differential Equations

Explicit and Implicit Methods

Here is the video of the eighth lecture that discusses the instability of Milne’s method, discusses the characterization of PDEs and begins the discussion of numerical solution of PDEs. We begin by discussing the parabolic PDEs, such as the heat equation using explicit numerical methods.

Topics to be covered in the next few lectures

6. Parabolic Differential Equations

Explicit and Implicit Methods

Initial and Boundary Conditions, Limiting Conditions

Convergence, Stability and Consistency

Well Posedness and Sufficiency

Lax-Wendroff Conservation Equations

Nonlinear Parabolic Equations and Schemes to Solve Them

Here is the video of the ninth lecture that discusses the numerical solution of the parabolic PDE characterized by the heat equation. We show the Neumann stability analysis for the explicit Forward Difference in time O(Dt, Dx²) scheme; the implicit Backward Difference in time O(Dt, Dx²) scheme; and we started discussing the Centered Difference in time O(Dt², Dx²) scheme. Here is a document related to the Neumann Stability analysis

Here is the video of the tenth lecture that discusses the numerical solution of the parabolic PDE characterized by the heat equation. We show the Neumann stability analysis for the Centered Difference in time O(Dt², Dx²) scheme and the problems related to that scheme; how the Crank-Nicholson Scheme O(Dt², Dx²) helped fix those problems. We also discuss the Lax Equivalence Theorem in the topic of Convergence, Stability and Consistency.

The eleventh lecture discusses multi-level time schemes and solution of 2-D heat equation using the ADI (alternating direction implicit) scheme, schemes for polar coordinate systems, and schemes for nonlinear parabolic equations

Also a take-home midterm was provided and due in class next Wednesday. Please do your own work.

The twelfth lecture discusses more information on cylindrical and polar problems for solutions related to the origin. Extrapolation to the limit is also discussed for the explicit method and for the Crank-Nicholson Schemes. Finally, a discussion of the Newton Linearization scheme for nonlinear parabolic equations is given.

The twelfth lecture covers a numerical problem for the Newton Linearization scheme, and looks at the Richtmeyer Linearization Scheme. A handout was given regarding elliptic PDEs which will be our next topic and their numerical solution methods.

7. Elliptic PDE's

Laplace Equation and Iterative Schemes to Solve It

Poisson Equation

Dealing with Limiting Conditions of Boundary Conditions

NonCartesian Meshes and NonRegular Regions

The thirteenth lecture covers Elliptic PDEs. Some basic theoretical discussions are given such as the principle of the maximum/minimum and numerical solution of the Laplace Equation Ñ²u = 0 and the Poisson’s Equation Ñ²u = f such as the Liebmann, Southwell and Gauss-Seidel methods are discussed. Proof of convergence of the iterative techniques is discussed to include choice of initial approximations to the solution is covered. Lastly, solution of the problem using the implicit technique is also discussed.

The fourteenth and fifteenth lectures continue to discuss elliptic PDEs

The sixteenth lecture covers the Laplace Equation Ñ²u = 0 and the Poisson’s Equation Ñ²u = f solved using the implicit method. We also discuss the general topic of handling non-standard boundaries, where distances from the standard grid for the interior region to the boundary leaves a portion of a stepsize. We also discuss grids in equilateral triangles and solution of the equations in such regions. We lastly show how the ADI method discussed for the 2-D heat equation can be used to solve the Laplace equation if the value of C = is taken to be very large. Next time we will discuss the hyperbolic PDEs and their solutions.

The seventeenth and eighteenth lectures continue to deal with parabolic type PDEs

8. Hyperbolic PDE's

Implicit and Explicit Schemes

Problems of Stability in the Schemes

Courant-Friedrichs-Levy Condition

Numerical Integration along Characteristics-maintaining discontinuities

Second Order Equations

The nineteenth lecture begins discussion of solution of hyperbolic PDEs such as the wave equation. The general finite difference equation is derived and its stability is discussed in which the Courant Number C = is an important factor. The domain of dependence and range of influence is defined and the numerical characteristics and the analytical characteristics are defined and related to each other and related to the stability of the numerical method as well as the accuracy of the numerical method. The solution for time level 1 is discussed using one of the initial conditions and a more accurate result is given using the D’Alembert Solution.

The twentieth and twenty-first lectures continue to discuss hyperbolic PDEs and their solutions.

The twenty-second lecture begins discussion of solution of hyperbolic PDEs using the method of characteristics. The equations to be solved are derived and two examples are discussed in detail. One example covers a situation in which initial conditions are discontinuous but the characteristics are constant. One example is where the characteristics are not constant but are only functions of the independent variables only, but the initial conditions are continuous.

The twenty-third lecture discusses the last example of the method of characteristics for a non-linear PDE in which the characteristics are a function of the dependent variable. We also discussed the one-dimensional wave equation and its solution.

The twenty-fourth lecture continues the discussion of the method of characteristics.

9. Optimization

Linear Programming

The twenty-fifth lecture and twenty-sixth lecture discuss the rudiments of linear programming, namely defining the feasible region, the constraint equations, solution of problems of two variables, solution of problems of multiple variables using the simplex method.

The twenty-seventh lecture will wrap up the course and cover the final exam.