Chapter01-Introduction.ppt

1、Computational Solid Mechanics计算固体力学,Department of Civil Engineering, Shanghai University,Definition What is a soild? it can also support a substantial shearing force over the time scale of some natural process or technological application of interest. Solidmechanics isconcernedwiththe stressing, def

2、ormationandfailureofsolidmaterialsandstructures,Computational Solid Mechanics,2,Objective of the course To provide the basic concepts, theories and methods of computational solid mechanics To mainly know well the theory and methods of the Finite Element Method, and to be able to derive the element a

3、nd global matrices of FEM To grasp the various numerical methods for dynamic/ static analysis of the structures with FME methods To understand the weighted residual method, finite difference method, finite volume method, boundary element method and meshless methods, etc. To provide a basis for using

4、 commercial FEM packages easily to solve practical problems,Computational Solid Mechanics,3,Computational Solid Mechanics,Contents 1 Introduction (引言) 2 Fundamentals for Mechanics of Solids and Structures (固体和结构力学基础) 3 Fundamentals for Finite Element Method (有限元方法基础) 4 FEM for Trusses (桁架的有限元方法) 5 F

5、EM for Beams (梁的有限元方法) 6 FEM for Frames (框架的有限元方法) 7 FEM for Two Dimensional Solids (二维固体有限元方法) 8 FEM for Plates and Shells (板壳有限元方法),4,Computational Solid Mechanics,Contents 9 FEM for Three Dimensional Solids (三维固体有限元方法) 10 Specialized Methods (专门化方法) 11 Modelling Techniques (建模技巧) 12 Finite Differ

6、ence Method (有限差分法) 13 Boundary Element Method (边界元方法) 14 Meshless Method (无网格方法),5,Chapter 1Introduction 引言,Chapter 1 Introduction,Contents 1.1 Computer-Aided Engineering (CAE) 1.2 Physical Problems in Engineering 1.3 Computational Modeling Using FEM 1.4 Simulation 1.5 Visualization 1.6 A Brief His

7、tory of the FEM 1.7 References,7,1.1 Computer-Aided Engineering (CAE),Chapter 1 Introduction,Computer-Aided Engineering (CAE) The use of computer software to solve engineering problems With the improvement of graphics displays, workstations, and graphics standards, computer-aided engineering (CAE) h

8、as come to mean the computer solution of engineering problems with the assistance of interactive computer graphics,9,Chapter 1 Introduction,Task of CAE: Modeling and simulations using computers in the design and analysis of a product Main work of CAE: Study deformations, stresses, heat transfer, flu

9、id flows, vibrations, noises, and so on, of components and systems CAE tools: FEM , FVM, BEM, MBD (Multiple body dynamics), FDM, etc. It can replace or complement most of the costly and time-consuming physical tests Have been applied in all science and engineering fields, and all industries,10,Chapt

10、er 1 Introduction,Benefit of CAE Boeing saved one test airplane ($100M) by using computer modeling and simulations in development of Boeing 777 The period for design a new car is reduced from 5-6 years to 1-2 years,Fast analysis of complex structures, such a high-rising building, etc. (Reliability,

11、Earthquake-proof, ),11,Chapter 1 Introduction,Applications of CAE Mechanical Engineering (ME),12,Chapter 1 Introduction,Applications of CAE Mechanical Engineering (ME),Simulation of motions of rigid bodies,Simulation of bucking of motorcar chassis,Simulation of car road,13,Chapter 1 Introduction,App

12、lications of CAE Mechanical Engineering (ME),Simulation of motions of a driver in traffic accident,Simulation of motion of mechanical arm,14,Chapter 1 Introduction,Applications of CAE Civil Engineering (CE),15,Chapter 1 Introduction,Applications of CAE Civil Engineering (CE),核废料深层处置关键问题,16,Chapter 1

13、 Introduction,Applications of CAE Aerospace Engineering (AE),17,Chapter 1 Introduction,Applications of CAE Electrical Engineering (EE),18,Chapter 1 Introduction,Applications of CAE Biomedical Engineering (BE),19,Chapter 1 Introduction,Current Practices in Engineering Ever increasing practices of CAD

14、/CAE/CAM/PLM (Engineering Product Lifecycle Management) Tools CAD (Computer-aided design) is a done greatly CAE are being used for 80-95% of analysis jobs to optimize the design and reduce the design cost and period Every aspect of engineering fields are covered by CAE (kinematics, dynamics, solid/f

15、luid mechanics, structural dynamics, noise controls, multi-physics phenomena, and large-scale simulations) CAM (Computer-aided manufacturing) /PLM still in intensive research, such as BIM in civil engineering,20,Chapter 1 Introduction,The Future of CAE CAD/CAE/CAM/PLM software will be widely and ine

16、xpensively available to every engineer Every engineer will be required to have the practical knowledge of CAE It will be able to analyze the problems of the large-scale, multi-scale, multi-physics, visual and instant Globalization of the CAE practices (e.g., jobs to India and China) Online computing

17、 - Intensive use of the new Internet,21,1.2 Physical Problems in Engineering,Chapter 1 Introduction,In building advanced engineering systems with CAE, engineers and designers go through a sophisticated process of modelling, simulation, visualization, analysis, designing, prototyping, testing, and la

18、stly, fabrication To ensure the workability, as well as for cost, much work is involved before the fabrication of the product This process is often iterative in nature, that is, some of the procedures are repeated based on the results obtained at a current stage Therefore, techniques related to mode

19、lling and simulation in a rapid and effective way play an increasingly important role,23,Chapter 1 Introduction,Process to fabrication of a product,24,Chapter 1 Introduction,Design process for an engineering system Major steps include computational modeling, simulation and analysis of results Proces

20、s is iterative Aided by good knowledge of computational modeling and simulation In modeling, it will be involved in various physical problems in engineering Mechanics for solids and structures Heat transfer Acoustics Fluid dynamic mechanics Others,25,Chapter 1 Introduction,The focus will be on the t

21、echniques of physical, mathematical and computational modelling, and various aspects of computational simulation A good understanding of these techniques plays an important role in building an advanced engineering system in a rapid and cost effective way Computational Solid Mechanics is strongly int

22、erdisciplinary, which consists of Theoretical and Applied Mechanics Pure and Applied Mathematics (Calculous, Algebra, ODE, PDE, and Integral-differential Equation, etc.) Computer and Information Sciences Numerical Analysis (Finite Element Method, Finite Difference Method, Finite Volume Method, Meshl

23、ess Method, etc.),26,Chapter 1 Introduction,The analysis process by computer methods includes three stages Mathematical modeling, discretization and solution Mathematical model has been subdivided into three broad classes Strong Form (SF) Weak Form (WF) Variational Form (VF),27,Chapter 1 Introductio

24、n,Computational Solid Mechanics,Computational Mechanics integrates aspects of four disciplines,The Finite Element Method (FEM) has developed into a key, indispensable technology in the modelling and simulation of advanced engineering systems in various fields like housing, transportation, communicat

25、ions, and so on,28,1.3 Computational Modeling Using FEM,Chapter 1 Introduction,The behaviour of a system depends upon the geometry or domain of the system, the property of the material or medium, and the boundary, initial and loading conditions For an engineering system, the geometry or domain, the

26、boundary and initial conditions can be complex It is, therefore, in general, very difficult to solve the governing differential equation via analytical means In practice, due to its practicality and versatility, most of the problems are solved using numerical methods. Among these, the methods of dom

27、ain discretization championed by the FEM are the most popular,30,Chapter 1 Introduction,The finite element method The FEM was first used to solve problems of stress analysis, and has since been applied to many other problems like thermal analysis, fluid flow analysis, piezoelectric analysis, and man

28、y others The FEM is a numerical method for seeking an approximate solution of the distribution of field variables in the problem domain that is difficult to obtain analytically It is done by dividing the problem domain into several elements, each of which usually has a very simple geometry,31,Chapte

29、r 1 Introduction,A continuous function of an unknown field variable is approximated using piecewise functions in each sub-domain, called an element formed by nodes The unknowns are then the discrete values of the field variable at the nodes Proper principles are followed to establish equations for t

30、he elements Then, the elements are tied to one another, which leads to a set of linear (nonlinear) algebraic/ differential equations for the entire system that can be solved easily to yield the required field variables,32,Chapter 1 Introduction,Computational simulation using FEM The procedure of com

31、putational simulation using the FEM software broadly consists of four steps Modeling of geometry Meshing (Discretization) Defining material properties Defining boundary, initial and loading conditions Some aspects of the above steps will be discussed briefly,33,Chapter 1 Introduction,Modeling of geo

32、metry Real structures, components are in general very complex, and have to be reduced to a manageable geometry Curved parts of the geometry and its boundary can be modeled using curves and curved surfaces, and the geometry is represented by a collection of elements The curves and curved surfaces are

33、 approximated by piecewise straight lines or flat surfaces, if linear elements are used,34,Chapter 1 Introduction,Figure shows an example of a curved boundary represented by the straight lines of the edges of triangular elements The accuracy of representation of the curved parts is controlled by the

34、 number of elements used,With more elements, the representation of the curved parts by straight edges would be smoother and more accurate The more elements, the longer the computational time that is required It is always necessary to limit the number of elements,35,Chapter 1 Introduction,Depending o

35、n the software package used, there are many ways to create a proper geometry for the FE mesh Points can be created simply by keying in the coordinates Lines/curves can be created by connecting points/nodes Surfaces can be created by connecting/rotating/ translating the existing lines/curves Solids c

36、an be created by connecting/rotating/ translating the existing surfaces Points, lines/curves, surfaces and solids can be translated/rotated/reflected to form new ones,36,Chapter 1 Introduction,Meshing (Discretization) Meshing is performed to discretize the geometry into small pieces called elements

37、or cells Usually, the solution of an problem is very complex, and varies in a way that is very unpredictable using functions across the whole domain of the problem If the problem domain can be divided (meshed) into small elements or cells, the solution within an element can be approximated easily us

38、ing simple functions, such as polynomials The solutions for all of the elements thus form the approximate solution for the whole problem domain,37,Chapter 1 Introduction,Proper theories are needed for discretizing the differential equations based on the discretized domains The theories used are diff

39、erent from problem to problem It can be a very time consuming task, and usually an experienced analyst will produce a more credible mesh for a complex problem The domain has to be meshed properly into elements of specific shapes Information must be created during the meshing for use later in the for

40、mation of the FEM equations,38,Chapter 1 Introduction,A complex function is represented by piecewise linear functions,39,Chapter 1 Introduction,Mesh generation is a very important task of the pre-process Automatic mesh generators: an ideal Semi-automatic mesh generators: in practice Shapes (types) o

41、f elements Triangular (2D) Quadrilateral (2D) Tetrahedral (3D) Hexahedral (3D) others,40,Chapter 1 Introduction,Mesh for the design of scaled model of aircraft for dynamic analysis,41,Chapter 1 Introduction,Mesh for a boom (起重臂) showing the stress distribution,42,Chapter 1 Introduction,Mesh of a hin

42、ge joint,43,Chapter 1 Introduction,Mesh of a driver and vehicle seat,44,45,Chapter 1 Introduction,Mesh of a slope,46,Chapter 1 Introduction,Mesh of a foundation pit excavation,Chapter 1 Introduction,Defining material properties Type of material property depends upon problem Usually involves simple k

43、eying in of data of material property in preprocessor Use of material database (commercially available) Experiments for accurate material property,47,Chapter 1 Introduction,Defining boundary, initial and loading conditions Very important for accurate simulation of engineering systems Usually involve

44、s the input of conditions with the aid of a graphical interface using preprocessors Can be applied to geometrical identities (points, lines/curves, surfaces, and solids) and mesh identities (elements or grids) Experience, knowledge and proper engineering judgments are required to accurately simulate

45、 these conditions,48,1.4 Simulation,Chapter 1 Introduction,Two major aspects when performing simulation Discrete system equations Principles for discretization Problem dependent Equations solvers Making use of computer architecture Problem dependent,50,Chapter 1 Introduction,Discrete system equation

46、s Based on the mesh generated, a set of discrete simultaneous system equations can be formulated using existing approaches There are a few types of approach for establishing the simultaneous equations Variational principles or Principle of virtual work, such as the Hamiltons principle, the potential

47、 energy principle, Generalized variational principles, and so on (FEM ) The weighted residual method, PDEs are satisfied in a weighted integral sense (FEM, EFM(element free method), FVM) The Taylor series (Finite Difference Method, FDM) The Conservation laws on each finite volume (elements) (Finite

48、Volume Method, FVM) ,51,Chapter 1 Introduction,Equations solvers After the discretized simultaneous equations have been created, it is fed to a solver to solve Two important considerations when choosing algorithms for solving a system of equations Required storage CPU (Central Processing Unit) time

49、needed There are two main types of methods for solving equations Direct methods Iterative methods,52,Chapter 1 Introduction,Direct methods (for small systems, up to 2D) Gauss elimination method LU decomposition method Central difference method Iterative methods (for large systems, 3D onwards) GaussJ

50、acobi method GaussSeidel method SOR (Successive Over-Relaxation) method Generalized conjugate residual method Line relaxation method Newmark method ,53,Chapter 1 Introduction,For nonlinear problems Another iterative loop is needed The nonlinear equation has to be properly formulated into a linear eq

51、uation in the iteration For time-dependent problems Time stepping is required First, solving for the solution at an initial time, then, using this solution to march forward for the solution at the next time step, and so on until the solution at the desired time is obtained There are two main approac

52、hes to time stepping Implicit approach (accurate but more computationally expensive) Explicit approach (simple, but less accurate),54,Chapter 1 Introduction,Available Commercial FEM Software Packages ANSYS (General propose, PC and workstations) ANDINA ( Fluid-solid coupled dynamics ananlysis) NASTRA

53、N (General purpose FEA) ABAQUS (Nonlinear and dynamic analysis) COSMOS (General purpose FEA) SAP2000 (General purpose FEA) PATRAN (Pre/Post Processor) HyperMesh (Pre/Post Processor) Dyna-3D (Crash/impact analysis) MIDAS (General purpose FEA) ,55,1.5 Visualization,Chapter 1 Introduction,The result is

54、 usually a vast volume of digital data These results are used to interpolate, analyze and present The visualization is performed ( post-processor) Methods of visualization In form of tables, text files and xy plots 3D object representation Wire-frames, Collection of elements, Collection of nodes 3D

55、object representation- rotate, translate, and zoom in/out Results: Contours, Iso-surfaces, Vector fields, fringes, deformations Visual reality A goggle, inversion desk, and immersion room,57,Chapter 1 Introduction,Visualization,58,Chapter 1 Introduction,Visualization,59,1.6 A Brief History of the FE

56、M,Chapter 1 Introduction,A Brief History of the FEM Physical intuition first brought finite element concepts, In the 1930s a structural engineer recognized that a truss was simply an assembly of rods 1943 - Courant used an assemblage of triangular elements and the principle of minimum potential energy to study the St. Venant torsion problem 1956 -Turner, Clough, Martin and Topp introduced direct stiffness method for plane stress problems by means of triangular elements, whose pr