CIEG 605 Intermediate Topics in Finite Element Analyses

Transcription

1 CIEG 605 Intermediate Topics in Finite Element Analyses 2016 Fall Semester General Description from Course Catalog Topics include finite element analyses of linear steady-state scalar field problems, linear elastostatics, analysis techniques for linear transient and dynamic analyses of solids, overview of elements commonly used in matrix structural analyses, and discussion of mixed and non-conforming finite element formulations. In each case, element formulations and solution algorithms are assessed, limitations and possible improvements are presented and numerical methods critically examined. Course Goals Building on the material presented in CIEG 601 (Introduction to the Finite Element Method), the following goals are associated with CIEG 605: To teach intermediate concepts associated with the finite element method. To develop valid mathematical models for physical problems. To use the finite element method in simulating various types of physical problems. To present important lessons to be learned when analyzing various classes of physical problems using the finite element method. To make the student aware of the importance of critically evaluating the results of approximate analyses in order to better assess their correctness. To make the student into a knowledgeable and critical numerical analyst. General Points Describing CIEG 605 CIEG 605 focuses on more specific problem areas related to approximate solutions using the finite element method. The emphasis in CIEG 605 is on linear problems in mechanics. The course consists of lectures, where basic ideas are developed and where modeling and programming techniques are discussed, and homework that supplements the lectures. CIEG Fall Semester

2 For their solution, a large percentage of the homework problems require the use of a digital computer and existing finite element software. These are used to obtain typical results that facilitate learning of the aforementioned lessons. Recommended Background The course draws on material from mechanics, numerical analysis, and linear algebra. As such, the student should have a strong knowledge of fundamentals associated with these subjects. Although not emphasized as heavily as the above subjects, aspects of computer science also are presented in select lectures. General Information Concerning Course Time Offered: 3:30 to 6:15 p.m. on Tuesdays and Thursdays. Duration: From 25 October to 8 December 2016 (i.e., the second 7-week portion of the 2016 fall semester). Room: 102 Colburn Lab Prerequisite: CIEG 601 or consent of instructor. introductory class in the finite element method. The student must have had an Instructor: V. N. Kaliakin, Professor office: TBA office hours: by or by appointment. voice: FAX : kaliakin@udel.edu www: Course Materials The following materials are associated with CIEG 605: Textbook: V. N. Kaliakin, Approximate Solution Techniques, Numerical Modeling and Finite Element Methods, New York: Marcel Dekker, Inc. (2001) (ISBN X), 674 pages. This is the textbook used in CIEG 401/601. In this class we quickly review certain aspects of Chapters 7 to 10. Chapters 11 and 12 are next covered in detail. In addition, certain appendices in the textbook are also discussed as appropriate. Supplementary Class Notes: These notes, in the form of PDF files, shall be made available during the semester as the associated subject matter is being discussed. CIEG Fall Semester

3 Some Potentially Useful Notes Selected Topics in Numerical Analysis. Each of the following PDF files contains background information related to a particular topic in numerical analysis that is pertinent to the finite element method. Taylor s Formula: Polynomials: Numerical Integration: Notes regarding vector algebra: Notes regarding linear algebra and matrices: Notes regarding coordinate transformations: Some References Associated with CIEG 605 K. J. Bathe, Finite Element Procedures, Prentice-Hall, New Jersey (1996). A. P. Boresi, R. J. Schmidt, and O. M. Sidebottom, Advanced Mechanics of Materials, 5th edition, John Wiley & Sons, New York, NY (1993). R. D. Cook, D. S. Malkus, M. E. Plesha and R. J. Witt, Concepts and Applications of Finite Element Analysis, 4th Edition, Wiley, New York (2002). T. J. R. Hughes, The Finite Element Method, Linear Static and Dynamic Finite Element Analysis, Prentice-Hall, New Jersey (1987). G. Strang and G. J. Fix, An Analysis of the Finite Element Method, Prentice-Hall, Inc., New Jersey (1973). L. J. Segerlind, Applied Finite Element Analysis, 2nd edition, John Wiley and Sons, New York, NY (1984). S. P. Timoshenko and J. N. Goodier, Theory of Elasticity, McGraw-Hill, New York (1970). CIEG Fall Semester

4 Course Information on World Wide Web Class Home Page Class Syllabus (PDF file) Lecture Summary (updated after every class lecture) List of Homework Assignments (updated every time new homework is assigned) Course Information on Sakai Class Syllabus (PDF file): Available from the Syllabus tab. Lecture Summary (updated after every class lecture): accessed using the following path from the course Sakai page: This links directly to Resources tab Contents of Lectures List of Homework Assignments (updated every time new homework is assigned): Available from the Assignments tab. Also available at Homework Solutions (updated after a graded homework assignment is returned): accessed using the following path from the course Sakai page: Resources tab Homework Solutions Selected Topics in Numerical Analysis: Each of the following PDF files contains background information related to a particular topic in numerical analysis that is pertinent to approximate solution techniques in general, and the finite element method in particular: CIEG Fall Semester

5 Notes on numerical integration Notes on polynomials Notes on Taylor s Formula These are accessed using the following path from the course Sakai page: Resources tab Numerical Analysis Notes Selected Topics in Mathematics: Each of the following PDF files contains background information related to a particular topic in mathematics. These are accessed using the following path from the course Sakai page: Notes regarding vector analysis Resources tab Math Notes Notes regarding linear algebra and matrices Notes on coordinate transformations PowerPoint files used in class: these files are accessed using the following path from the course Sakai page: Resources tab PowerPoint files Supplementary Lecture Materials: these materials, which are typically provided as PDF files, are accessed using the following path from the course Sakai page: Resources tab Supplementary Lecture Materials (PDF files) CIEG Fall Semester

6 Course Content The topics listed below shall be discussed in class lectures. All topics do not receive the same lecture time some are discussed in detail, while others only briefly. Pertinent notes, in the form of PDF files, shall be provided as needed. 25 October 2016: Review of some fundamental of the finite element method. General approach for developing finite element equations and their convergence [Chapter 7]. Fundamental properties of element interpolation functions [Chapter 9]. Overview of fundamental aspects of element mapping [Chapter 10]. Linear steady state scalar field problems [Chapter 11]. Examples of physical applications (boundary value problems) in this problem class. Key theoretical aspects. General element formulations. 27 October 2016: Torsion of prismatic, non-circular cross-sections [Chapter 11]. Strong form of the problem. Element equations. Development of valid mathematical models and results of sample analyses. 1 November 2016: Finite element analyses in linear elastostatics [Chapter 12]. Review of specific elastic idealizations and their associated general finite element equations: Brief overview of Theory of Elasticity 1. Development of general element equations. Insight into the structure and contents of the C matrix. Specific element examples. Three-dimensional analyses. Plane stress idealizations. Generalized plane strain and plane strain idealizations. Axisymmetric idealizations (Special cases 1 and 2). 1 The study of elastostatic problems includes a brief overview of the Theory of Elasticity; consequently, a portion of this course complements MEEG 610 (Intermediate Solid Mechanics). CIEG Fall Semester

7 3 November 2016: Concentrated nodal specifications. Computation of equivalent nodal forces from distributed loading. Specific examples in linear elastostatics and their finite element solution [Supplementary Notes (PDF)]. Examples involving compressible materials. Behavior in the incompressible limit ( volumetric locking). 8 November 2016: Election Day no lecture. 10 November 2016: Finite element analysis of beam axial-flexural response [Supplementary Notes (PDF)]. General remarks. Problem class under consideration. Classes of exact solutions. Element types used. Simulation of beam bending using irreducible continuum elements. Examples Insight into shear deformations. Element Limitations: why some elements give very poor results (due to shear locking ). Discussion of potential remedies. 15 November 2016: Specialized ( structural ) elements in solid mechanics (e.g., axial ( truss ), torsion, beam and frame elements) [Chapter 13 of Supplementary Notes (PDF)]. Bernoulli-Euler versus Timoshenko two-dimensional beam elements and the issue of shear deformation. Limitations of three-dimensional Bernoulli-Euler beam and frame elements. 17 November 2016: Non-conforming elements [Supplementary Notes (PDF)]. General aspects of the formulation and comparison with conforming elements [Chapter 7]. Specific examples of non-conforming elements applied to beam bending and response in the incompressible limit. CIEG Fall Semester

8 22 and 24 November 2016: Thanksgiving holiday no lecture. 29 November 2016: Analysis techniques for linear transient analyses (i.e., those that are first-order in time) [Chapter 14 of Supplementary Notes (PDF)]. General governing equation; boundary and initial condition. Solution algorithms. Algorithmic stability and oscillations. Sample analyses. 1 December 2016: Analysis techniques for linear dynamic analyses (i.e., those that are second-order in time) [Chapter 14 of Supplementary Notes (PDF)]. General governing equation; boundary and initial condition. Solution algorithms. Algorithmic stability and oscillations. Discussion of mass and damping matrices. 6 December 2016: Mixed formulations and associated elements [Chapter 16 of Supplementary Notes (PDF)]. Definition of irreducible and mixed formulations. Types of physical problems employing a mixed formulation. Volumetric locking. Elements with discontinuous pressure approximations and assessment of their stability via the heuristic engineering patch test. Example problems. 8 December 2016: Discussion of any loose ends from previous lectures. Mixed formulations and associated elements [Chapter 16 of Supplementary Notes (PDF)]. Elements with continuous pressure approximations and assessment of their stability via the heuristic engineering patch test. Example problems. CIEG Fall Semester

9 Grading The course grade consists of the following: Homework (including computer exercises) [70%]. Term Project (due Friday, December 9, 2016) [30%] 2 Academic Honesty Quoting the Code of Conduct: All students must be honest and forthright in their academic studies. To falsify the results of ones research, to steal the words or ideas of another, to cheat on an assignment, or to allow or assist another to commit these acts corrupts the educational process. Students are expected to do their own work and neither give nor receive unauthorized assistance. Further details are available at the following URL: Policy for Listeners In order to receive a grade of L, listeners must attend all lectures. The only exceptions are excused (for legitimate reasons, of course) absences. If the number of available seats in the classroom is limited, priority will go to students desiring to take the class for credit. Cell Phone Policy Interruptions of class due to ringing cellular phones are unacceptable. Students possessing such phones must thus turn off the ringer (if not the phone altogether) before entering class. 2 The research papers must be written on an appropriate topic that is of interest to the student. Prior to beginning work on the research papers, the chosen topics must be discussed with the instructor (additional details concerning the respective research papers shall be given at a later date). CIEG Fall Semester

10 Potential Term Project Topics Project Description: due no later than 5:00 p.m. on Friday, October 28, Completed Project: due no later than 5:00 p.m. on Friday, December 9, As noted above, the term project consists of either an extensive term paper on a topic of interest to the student or a significant programming exercise. In either case, prior to beginning work on it, the chosen project must be discussed with the instructor. I therefore ask that you please submit to me a description of your proposed term project. After reviewing these descriptions, I will discuss the scope the proposed projects with each one of you separately. Listed below are some possible topics for the term project many more can, however, be dreamed up. Research Papers Investigate, in a fairly broad sense, the p- or h-p forms of the finite element method. Investigate error estimates and adaptive techniques used in conjunction with finite element analyses; investigate the general aspects of this topic but focus on a specific application. Investigate some aspect related to the numerical simulation of incompressible or nearly incompressible materials. Investigate numerical schemes used in elastodynamic finite element analyses (emphasis can be on a particular scheme, its implementation, etc.). Study approaches used in simulating particular classes of composite materials. Investigate stress smoothing/projection techniques. Investigate so-called infinite elements in general, or with respect to some specific class of elements (e.g., mixed elements). Investigate hybrid elements (e.g., Pian s pioneering work). Investigate some application of non-conforming elements (e.g., bending simulations, behavior in the incompressible limit, etc.). Study the finite element solution for a given problem class from the Theory of Elasticity and compare to closed form solutions. 3 Please submit the project proposal electronically as an attachment. CIEG Fall Semester

11 Programming/Analysis Exercises Develop, implement and test a higher-order 1-d beam element (e.g., a three-node Timoshenko beam element that shall be used in conjunction with bi-quadratic continuum elements; a three-dimensional Bernoulli-Euler beam element, etc). Develop a higher-order element (e.g., a 15-node triangle). Implement the element in an existing finite element computer program. Verify the element s performance through the analysis of various simple problems beginning, of course, with the engineering patch test. Extend a steady state 2-d heat conduction code to one for transient response. Develop, implement and test a curved beam element. Perform a thorough analysis of standard one-dimensional beam elements (Bernoulli- Euler and/or Timoshenko) and compare with standard beam theory solutions as well as with the theory of elasticity. Develop, implement and test higher-order continuum elements for scalar field problems. Perform a parametric study of selected flow through porous media problems (e.g., vary the permeability, layer geometry, extent of sheet piles, etc.). Perform a parametric study of a class of problems (classical or new) from Geotechnical Engineering. Examples include loads applied to finite and semi-infinite solution domains. Investigate the models, element formulations and analyses performed (e.g., simple foundations) using beam on elastic foundation elements. CIEG Fall Semester

12 Guidelines for Research Papers The paper must be written using word processing software (e.g., Microsoft Word, L A TEX, etc.). The paper must contain an introduction, a body of text, and a conclusion. The paper must have a consistent document style. Where appropriate, equations must be numbered; all variables, constants, etc. must be defined. All figures must be easily legible and must have captions. All tables must be easy to understand and must have captions. Proper citations must be given in the body of work. A complete list of references must be provided. The style used in this list must be consistent. CIEG Fall Semester