Beijing University of Chemical Technology
Syllabus for The Finite Element Method and Numerical Analysis
Ⅰ. General Information
Course Code |
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Course Information | Academic Discipline |
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Total Class Hours | 40 | Credits | 2.5 | Lecture Hours | 32 | Laboratory Hours | 0 | Computer Lab Hours | 8 |
Course Title (in Chinese) | 工程有限元法与数值分析 | ||||||||
Course Title (in English) | The Finite Element Method and Numerical Analysis | ||||||||
Applicable Majors | mechanical engineering and automation | ||||||||
Semester Available | 8 | ||||||||
Prerequisites (Course Title) | Advanced mathematics, material mechanics, elasticity, linear algebra | ||||||||
Corequisites (Course Title) |
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Brief Course Description | This course is a basic course of technology for undergraduates in engineering majors. The finite element method (FEM) is an effective numerical method to solve the partial differential with initial and boundary conditions, widely applied in the science research and engineering analysis. Based on the elastic mechanics and fluid mechanics, the fundamental theory and the technical route of the FEM are introduced, including the basic route of FEM, the FEM for bar structural problems, the FEM for plane problems, the FEM for space axisymmetric problems, the FEM for space problems in Elasticity, the FEM for Newtonian fluid flow, the introduction of common elements and so on. To establish and solve the FEM equations, the numerical analysis methods such as the function interpolation, numerical differentiation and integration, equation roots, etc. and the corresponding computer programming method are introduced. Develop the ability to solve the typical load problems of structures and computational fluid dynamics with programed procedures by studying, programming and using the codes for the finite element analysis during the computer practice class. After studying this course, the students could master the basic idea and analysis steps of the FEM, how to establish the finite element model, and the numerical analysis method for FEM programming, as well as how to solve simple engineering problems with procedures for the finite element analysis. That would lay a good foundation for students to solve the practical complicated engineering problems and improve their abilities to analyze and deal with these problems. | ||||||||
Ⅱ.Curriculum Nature and Course Objectives
2.1 course nature
Engineering finite element method and numerical analysis is a theoretical and comprehensive technical basic course. It is mutually verified and supported with theoretical analysis and experimental analysis methods. It is a professional elective course for the major of mechanical design, manufacturing and automation in Colleges and universities.
2.2 course objectives
By teaching the basic idea of finite element method, analysis steps, establishment of finite element model and analysis of typical engineering problems through finite element program, cultivate students' ability to comprehensively use their knowledge to analyze and solve engineering practical problems, and lay a foundation for students to further study the core courses of mechanical specialty and complete their graduation design thesis. Specific course objectives are as follows:
G1: be able to use finite element method to establish and solve mathematical models for typical machining processes and typical structures, so that students can have the ability of numerical analysis;
G2: enable students to master the method of analyzing engineering problems by using finite element method, and cultivate students to comprehensively apply their knowledge to analyze simple fluid flow problems and structural mechanics problems by different methods, so as to lay a foundation for analyzing complex problems;
G3: enable students to master the programming method of finite element method, and cultivate students' ability to use software to analyze and solve practical engineering problems.
Ⅲ. The Corresponding Relationship between Course Objectives and Graduation Requirements
Table 1 corresponding relationship and supporting weight between curriculum objectives and graduation requirements index points
Grade I index points required for graduation | Secondary index points for graduation requirements | Course objectives | Approach to achievement | Evaluation basis | Degree of support (h, m, l)
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1 engineering knowledge | 1-2 be able to establish and solve mathematical models for research objects in mechanical design, manufacturing and control. | G1 | Classroom performance, homework, examination | In class tests, assignments, examination
| M |
2 problem analysis | 2-2 be able to correctly express complex engineering problems in mechanical design, manufacturing and control based on relevant scientific principles and mathematical model methods. | G2 | Classroom performance, homework, examination | In class tests, assignments, examination
| H |
5 use modern tools | 5-2 be able to select and use appropriate instruments, information resources, engineering tools and professional simulation software to analyze, calculate and design engineering problems in mechanical design, manufacturing and control. | G3 | Classroom performance, homework, Computer operation,examination | In class tests, assignments, computer opertion , examination
| H |
Ⅳ. Teaching Contents and Requirements for the Lecturing Part
Part I: finite element method of structural mechanics
4.1 introduction and formulation of elasticity problems (3 class hours)
4.1.1 teaching objectives (G1, G2)
Clarify the status, function and development of the course, and understand the nature, learning methods and requirements of the course.
4.1.2 teaching content
(1) The basic idea of finite element method;
(2) The occurrence and development of finite element;
(3) Course learning methods;
(4) General analysis steps of finite element method;
(5) Basic knowledge of elasticity.
4.1.3 teaching requirements
Understand the essence, technical route and learning method of finite element; Master the basic concepts, basic variables and mathematical formulation of elasticity; Understand the relationship between finite element method, experimental mechanics and other mechanical theoretical methods.
4.2 triangular element analysis (3 class hours)
4.2.1 teaching objectives ( G2)
Master the specific implementation process of discretization and element characteristic matrix establishment in the analysis steps of finite element method.
4.2.2 teaching content
(1) Discretization of finite element method;
(2) The establishment of element stiffness matrix of triangular element;
(3) The element load matrix of triangular element is established.
4.2.3 teaching requirements
Master the concept and content of discretization: point number and point coordinates, element node information, boundary discretization, equivalent treatment of external forces; Deeply understand the pseudo displacement mode in the element, express the undetermined parameters with node displacement, the expression and properties of shape function, the strain matrix in the element, the stress matrix in the element, the element stiffness matrix K, and the calculation formula of equivalent node force; Be familiar with the properties of single stiffness and the steps of calculating single stiffness and equivalent nodal force.
4.3 overall analysis of triangular element (3 class hours)
4.3.1 teaching objectives (G2, G3)
Master the specific implementation process of integrating element stiffness matrix and element equivalent node force array to form structural stiffness matrix and structural node load array, and understand the establishment of finite element equation.
4.3.2 teaching content
(1) The finite element equation is established by the node balance method;
(2) The finite element equation is established by using the principle of virtual work;
(3) The integration of structural stiffness matrix and structural node load array;
(4) Storage of structural stiffness matrix;
(5) The introduction of displacement constraints.
4.3.3 teaching requirements
Master the specific technology of forming the overall balance equations, the diagonal 1 method and the diagonal element multiplication large number method; Understand the node equilibrium method, establish the finite element equation and delete row and column displacement constraint introduction method; Familiar with the properties and storage of structural stiffness matrix.
4.4 finite element solution (3 class hours)
4.4.1 teaching objectives (G2, G3)
Master the factors affecting the accuracy of finite element numerical solution and the general process of finite element method solution, and be able to understand and compile the finite element program for triangular element plane problems.
4.4.2 teaching content
(1) Solutions of algebraic equations;
(2) Calculation of other relevant quantities;
(3) Convergence and accuracy of the solution;
(4) The general process of finite element method for continuum elasticity analysis.
4.4.3 teaching requirements
Master the convergence criterion of finite element, the solution and iteration of finite element equation; Deeply understand the causes affecting accuracy and stress calculation and post-processing, and be familiar with the general process of finite element method for continuum elastic analysis; Learn to compile finite element solution program.
Part II: finite element method of computational fluid dynamics (12 class hours)
4.5 mathematical description of fluid problems
4.5.1 teaching objectives (G1, G2) (1 class hour)
Understand the basic composition of Newtonian fluid flow control equations and the composition of boundary conditions of flow problems; Master the overall idea of using finite element method to solve Newtonian fluid flow problems.
4.5.2 teaching content
(1) Finite element method is the core step of solving fluid flow problems;
(2) The basic composition of the governing equations of Newtonian fluid flow and the composition of the boundary conditions of the flow problem;
4.5.3 teaching requirements
Master the overall idea of using finite element method to solve Newtonian fluid flow problems; Understand the objectives of the core steps.
4.6 finite element mesh discretization, data storage and interpolation function transformation
4.6.1 teaching objectives (G2, G3) (2 class hours)
Understand the basic idea of calculation area discretization and master the storage format of finite element discretization data; Understand the interpolation function of finite element method and the establishment method of derivative relationship between global coordinates and local coordinates;
4.6.2 teaching content
(1) The basic idea of calculating regional discretization and the storage format of finite element discrete data;
(2) Composition and storage format of JM, jxy, JB1, jb2 and other data.
(3) Master the characteristics of interpolation function and the mapping relationship of coordinate system based on interpolation function.
4.6.3 teaching requirements
(1) It can construct the stored data according to the discrete node and element data; (2) Be able to understand the format of core data storage; (3) Master the transformation relationship between local coordinates and overall coordinates.
4.7 construction of weighted residual equation
4.7.1 teaching objectives (G2, G3) (3 class hours)
Master the establishment method of continuity equation and motion equation weighted residual equation
4.7.2 teaching content
(1) The establishment method of continuity equation and motion equation weighted residual equation.
(2) Application of distribution integral and Green's formula in derivative reduction.
4.7.3 teaching requirements
(1) The weighted residual equation of continuity equation can be constructed; (2) The weighted residual equation of continuous motion equation can be constructed.
4.8 establishment of element equation
4.8.1 teaching objectives (G2, G3) (3 class hours)
Master the integration region conversion idea of constructing element equation from weighted residual equation, and master the solution process of CE, De, be and Fe element equation sub blocks.
4.8.2 teaching content
(1) The integration region transformation idea of constructing the element equation from the weighted residual equation, and the solution process of the sub block of CE, De, be and Fe element equation;
(2) Preparation of matlab calculation program for each sub block.
4.8.3 teaching requirements
(1) Master the solution process of each sub block of internal element equation. (2) Master the solution process of each sub block of boundary element equation. (3) Be able to read the calculation program of each sub block in combination with the flow chart.
4.9 assembly of the overall method and substitution of boundary conditions for solution
4.9.1 teaching objectives (G2, G3) (3 class hours)
Understand the basic method of combining global equation subblocks by element equation subblocks; Understand the method of bringing in velocity boundary conditions; Understand the method of equation solving and result splitting using MATLAB.
4.9.2 teaching content
(1) The basic method of combining global equation subblocks from element equation subblocks;
(2) The method of introducing velocity boundary conditions;
(3) Using MATLAB to solve the equation and split the results.
(4) Based on the calculation results, the flow field characteristics are analyzed.
4.9.3 teaching requirements
(1) Be able to master the method of assembling the overall equation matrix from the element equation matrix; (2) Master the application method of velocity boundary conditions; (3) Be able to read the assembly program, boundary condition substitution program and equation solving program of each sub block in combination with the flow chart. (4) It can extract the calculation results of self-made program and analyze the results.
Part III: the numerical analysis related to the finite element method
4.10 introduction and error of numerical analysis (2 class hours)
4.10.1 teaching objectives (G1)
Understand the main sources of errors in numerical calculation and analysis, master the basic methods of numerical calculation error analysis, master some criteria in numerical calculation, and avoid iterative divergence and numerical calculation error accumulation.
4.10.2 teaching content
(1) The basic concept of error;
(2) Sources and influencing factors of errors;
(3) Some criteria in numerical operation.
4.10.3 teaching requirements
Master the influence of computer numerical storage accuracy and error transmission on numerical calculation results in numerical analysis, master several criteria in numerical operation, and be able to design corresponding numerical calculation programs in computer language to verify the relevant criteria of numerical calculation.
4.11 numerical differentiation and numerical integration (2 class hours)
4.11.1 teaching objectives (G1)
Understand the basic idea and principle of numerical integration, and be able to use Gaussian integral formula to numerically integrate the shape function in the process of finite element analysis; Master the basic methods of numerical differentiation.
4.11.2 teaching content
(1) Numerical differentiation;
(2) The basic idea of numerical quadrature;
(3) Equidistant node quadrature formula (Newton Cotes);
(4) Gaussian quadrature formula.
4.11.3 teaching requirements
Master the basic principle of equal node quadrature formula, and be able to select a reasonable quadrature formula according to the order of the integrand function; Focus on mastering Gaussian integral method, be able to use undetermined coefficient method and orthogonal polynomial to construct Gaussian formula, and be able to compile corresponding computer programs to realize numerical integration and numerical differentiation of functions.
4.12 interpolation and approximation (2 class hours)
4.12.1 teaching objectives (G1)
Be familiar with the basic concept of interpolation, master the basic methods and principles of conventional interpolation methods, master the least square method of curve fitting, and be able to properly interpolate and fit scattered data.
4.12.2 teaching content
(1) Interpolation concept;
(2) Lagrange interpolation (interpolation formula and remainder);
(3) Newton interpolation (mean difference, interpolation formula and remainder);
(4) Piecewise interpolation (piecewise linear and spline interpolation);
(5) Least square method of curve fitting.
4.12.3 teaching requirements
Understand the basic principles of Lagrange interpolation, Newton interpolation and piecewise interpolation methods, be able to select appropriate interpolation methods to interpolate scatter points according to data characteristics, master the basic principle of least square method of curve fitting, and learn to use computer language for data point interpolation and curve fitting.
4.13 root of equation (3 class hours)
4.13.1 teaching objectives (G1)
Master the basic idea of solving approximate roots of equations by iterative method, and understand the basic principle of Newton's iteration; Master the solution method of linear equations in finite element equations, and be able to use Gaussian elimination method to solve linear equations; Master the solution method of second-order differential equations with constant coefficients, and be able to use Newmark- β The dynamic response of finite element equation is calculated by finite element method.
4.13.2 teaching content
(1) Newton method;
(2) Gauss elimination method;
(3) Newmark- β Law;
(4) Solve the univariate equation;
(5) Solving linear equations;
(6) Solve the second-order differential equations with constant coefficients.
4.13.3 teaching requirements
Understand the derivation process of Newton iterative formula and learn to use Newton method to solve the approximate root of the equation; The Gauss elimination method and Newton iteration method can be compiled by computer language to solve the node displacement array coefficient in the finite element equation under static force; Can use Newmark- β Methods the transient vibration displacement response of finite element dynamic equations (second-order constant coefficient differential equations) is solved.
Ⅴ.Teaching Contents and Requirements for the Practical Part
5.1 teaching objectives (G3)
Master the preparation of finite element method program and the use of general finite element software ANSYS to carry out finite element numerical analysis of structural stress problems and hydrodynamic problems.
5.2 teaching content
Part I: finite element method of structural mechanics
(1) Computer debugging and calculation of plane problem program of triangular element (1 class hour);
(2) Use ANSYS software to calculate plane problems and member structure problems (1 class hour);
(3) Use ANSYS software to calculate general structure problems (2 class hours);
Part II: finite element method of computational fluid dynamics
(1) Discrete data extraction based on gambit software meshing results; (2 class hours);
(2) Use and result analysis of self-made fluid calculation program in class (2 class hours).
5.3 teaching requirements
Master the programming idea and solution method of finite element method program; Master the interface and analysis steps of general finite element software ANSYS to analyze structural problems; Master the method of compiling the core calculation program of hydrodynamics finite element; Familiar with the writing of analysis report; Be able to use the theoretical knowledge to analyze the numerical results.
Ⅵ. Evaluation Standards
There are about 4 class quizzes, each time about 10 minutes, without notifying students in advance. The final exam lasts 120 minutes. The specific examination time shall be notified to students at least 2 weeks in advance. The answer must be completed independently. Any cheating will result in being submitted to the school's student disciplinary committee for handling.
Table 2 assessment methods, contents and proportion of courses
Assessment method | Assessment method | Proportion | Main assessment contents |
Process assessment（40%） | Attendance and in class quiz | 10% | Class situation |
Homework after class | 20% | Completion of homework after class | |
Computer operation | 10% | Completion of computer operation | |
Result assessment （60%） | final exam | 60% | Overall assessment content |
Table 3 course assessment methods and contents and their supporting relationship to graduation requirement index points
Graduation requirement index point | Course objectives | Score | Assessment method
| Proportion | Main assessment contents |
1.2 be able to establish and solve mathematical models for research objects in mechanical design, mechanical manufacturing and control in the field of mechanical engineering. 2.2 be able to correctly express complex engineering problems in mechanical design, mechanical manufacturing and control in the field of mechanical engineering based on relevant scientific principles and mathematical model methods. | G1 | 20 | Classroom discussion | 10% | Attendance, questions, etc |
Exercise assignment | 20% | Completion of homework after class | |||
final exam | 70% | Teaching content supporting course objective G1 | |||
Graduation requirement index point 1.2 be able to establish and solve mathematical models for research objects in mechanical design, mechanical manufacturing and control in the field of mechanical engineering. 2.2 be able to correctly express complex engineering problems in mechanical design, mechanical manufacturing and control in the field of mechanical engineering based on relevant scientific principles and mathematical model methods. | G2 | 40 | Classroom discussion | 10% | Attendance, questions, etc |
Exercise assignment | 20% | Completion of homework after class | |||
final exam | 70% | The teaching content supporting the course objective G2 | |||
5.2 be able to select and use appropriate instruments, information resources, engineering tools and professional simulation software to analyze, calculate and design engineering problems such as mechanical design, mechanical manufacturing and control in the field of mechanical engineering. | G3 | 40 | Classroom discussion | 10% | Attendance, questions, etc |
Exercise assignment | 20% | Completion of homework after class | |||
operate a computer | 25% | Completion of computer case | |||
final exam | 45% | Teaching content supporting course objective G3 |
6.2 scoring criteria
Table 4 scoring criteria for classroom performance
Assessment index | weight | 100-90 | 89-80 | 79-70 | 69-60 | 59-0 |
Class attendance | 0.50 | 90% and above | 80% and above | 70% and above | 60% and above | absent |
In class test | 0.50 | Correct answer | Mostly correct | Basically correct | Partially correct | Wrong question
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Table 5 scoring criteria for exercises
Assessment index | weight | 100-90 | 89-80 | 79-70 | 69-60 | 59-0 |
Punctuality of completion | 0.1 | on time | on time | on time | Not on time | Not on time |
accuracy | 0.70 | Complete, high accuracy | Complete, high accuracy | Completion, average accuracy | Completed, many errors | Outstanding homework
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Clarity of expression | 0.20 | Neat writing | Good writing recognition
| Average writing recognition | Poor writing recognition | Can not be understood
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Table 6 scoring criteria for computer operation
Assessment index | weight | 100-90 | 89-80 | 79-70 | 69-60 | 59-0 |
Punctuality of completion | 0.5 | The report is comprehensive | The main steps in the report are complete | The report describes most of the main steps | Incomplete core steps | There is no process description in the report
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accuracy | 0.5 | The result is positive and the analysis is reasonable | The calculation results can be obtained, and the result analysis is basically reasonable | Some results can be analyzed | Results, lack of analysis | No result, incorrect analysis |
Ⅶ.Textbooks and Recommended References
7.1 teaching materials
[1] Leng Jitong, Zhao Jun, Zhang Ya. Fundamentals of finite element technology. Beijing: Chemical Industry Press, 2007
7.2 reference books
[1] Wang Xucheng. Finite element method. Beijing: Tsinghua publishing house, 2004
[2] Tirupathi R. chandrupatla, Ashok D. belegundu. Translated by Zeng pan and Lei Liping. Finite element method in Engineering (4th Edition). Beijing: Machinery Industry Press, 2014
[3] Klaus-Jurgen Bathe．Finite Element Procedures. Prentice Hall, Inc, 1996.
[4] Jiang Jianjing, he Fanglong, he Yibin, et al. Finite element method and its application. Beijing: Machinery Industry Press, 2015
[5] Steven C Chapra. Applied Numerical methods with MATLAB for Engineers and Scientists (3rd). McGraw-Hill, 2010.
Outline written by: Bi Chao
Course leader: Wang Weimin
Lecturers: Bi Chao, Zuo Yanfei, Liu Ning
Reviewed by: Ma Xiuqing
Discipline leader: Ma Xiuqing