同济大学：《有限元方法》课程教学资源（教案讲义）Chapter 1 Finite Element Method - Introduction to Finite Element Method

2014/5/15 Definition of FEM low does FEA work? Exple F=∫k2+6h Duer odmos:F12.667 ￥1y=x2+6 yy=x2+6 +6 The the area e curve How does FEA work? How ders FEA work? Integration using nmerica method ch y=x2+6 y=x2+6 V+y-I+6 ，y=2+6 As the number of sections inereases the error decrease Different functionscan be chosen toapproximate() 1

2014/5/15 1 FEM – Chapter 1 Introduction to Finite Element Method Definition of FEM  Many problems in engineering and applied science are governed by differential equations.  Few problems can have an exact, closed-form solution.  However, complexities in the geometry, properties and in the boundary conditions that are seen in most real-world problems usually means that an exact solution cannot be obtained.  The FEM is a numerical procedure for obtaining approximate solutions to many of the problems encountered in engineering analysis. FEM Computer implementation • Preprocessing(build FEM model, including geometry, loads, constraints, materials, element types, element connectivity (meshing), etc., may use CAD modeling) • Solver (assemble and solve the system of equations; As it is not uncommon for a finite element model to be represented by tens of thousands of equations, special solution techniques are used to reduce data storage requirements and computation time) • Postprocessing ( display and edit the calculation results, e.g. plot, color-coded plot, animation, etc.)

2014/5/15 How dees FEA work? How does FEA work? Twe key steps: in the domai )no ncde 正

2014/5/15 2 (a) A general two-dimensional domain of field variable (x, y). (b) A three-node finite element defined in the domain. (c) Additional elements showing a partial finite element mesh of the domain. (a) Arbitrary curved-boundary domain modeled using square elements. Stippled areas are not included in the model. A total of 41 elements is shown. (b) Refined finite element mesh showing reduction of the area not included in the model. A total of 192 elements is shown. (a) Tapered circular cylinder subjected to tensile loading: r(x)= r0- (x/L）/(r0- rL). (b) Tapered cylinder as a single axial (bar) element using an average area. Actual tapered cylinder is shown as dashed lines. (c) Tapered cylinder modeled as two, equal-length, finite elements. The area of each element is average over the respective tapered cylinder length. (d) Tapered circular cylinder modeled as four, equal-length finite elements. The areas are average over the respective length of cylinder (element length= L/4). (a) Displacement at x=L for tapered cylinder in tension. (b) Comparison of the exact solution and the four-element solution for a tapered cylinder in tension

2014/5/15 ustries,and even students rimary unknowns Pre-processing Fun What is this class about? Physical model FEA model Physical model FEA model FEA theory Results FEA theory Poat-processing Fun! FEA core Math!! 3

2014/5/15 3 Comparison of the computed axial stress value in a tapered cylinder.  In the FEM, a complex region defining a continuum is discretized into simple geometric shapes called elements.  The properties and the governing relationships are assumed over these elements and expressed mathematically in terms of unknown values at specific points in the elements called nodes.  An assembly process is used to link the individual elements to the given system. When the effects of loads and boundary conditions are considered, a set of linear or nonlinear algebraic equations is usually obtained.  Solution of these equations gives the approximate behavior of the continuum or system.  The continuum has an infinite number of degrees-of-freedom (DOF), while the discretized model has a finite number of DOF. This is the origin of the name, finite element method.  The number of equations is usually rather large for most real￾world applications of the FEM, and requires the computational power of the digital computer. The FEM has little practical value if the digital computer were not available.  Advances in and ready availability of computers and software has brought the FEM within reach of engineers working in small industries, and even students. Two features of the finite element method are worth noting.  The piecewise approximation of the physical field (continuum) on finite elements provides good precision even with simple approximating functions. Simply increasing the number of elements can achieve increasing precision.  The locality of the approximation leads to sparse equation systems for a discretized problem. This helps to ease the solution of problems having very large numbers of nodal unknowns. It is not uncommon today to solve systems containing a million primary unknowns

2014/5/15 Objectives of this FEM course: Areas focused on in this course .Understand the fundamental ideas of FEM: Static Dynamic .Can interpret and evaluate the quallty of the results Be aware of the limitations of FEM. Non-linea FEM in Structure analysis-the procedures sner bsmelmgngdngBaeumsndehangeue oingthngodes (e Origins of the Finite Element Method Origins of the Finite Element Method 4

2014/5/15 4 Objectives of this FEM course: • Understand the fundamental ideas of FEM; • Know the behavior and usage of each type of elements covered in this course; • Be able to prepare a suitable FEM model for given problems; • Can interpret and evaluate the quality of the results; • Be aware of the limitations of FEM. Static Dynamic Linear A little if time is allowed Non-linear Areas focused on in this course Linear static analysis Most structural analysis problems can be treated as linear static problems, based on the following assumptions: • Small deformation (loading pattern is not changed due to the deformed shape) • Elastic materials (no plasticity or failures) • Static loads (the load is applied to the structure in a slow or steady fashion) Linear analysis can provide most of the information about the behavior of a structure, and can be a good approximation to many analyses. It is also the bases of nonlinear analysis in most of the cases. FEM in Structure analysis – the procedures • Divide the structure into pieces (elements with nodes and DOFs) • Describe the behaviors of physical quantities on each element • Connect (Assemble) the elements at nodes to form an approximate system of equations for the whole structure • Solving the equations involving unknown quantities at nodes (e.g. displacement) • Calculate the desirable quantities (e.g. strain and stress) Origins of the Finite Element Method • It is difficult to document the exact origin of the FEM, because the basic concepts have evolved over a period of 150 or more years. • The term finite element was first coined by Clough in 1960. In the early 1960s, engineers used the method for approximate solution of problems in stress analysis, fluid flow, heat transfer, and other areas. • The first book on the FEM by Zienkiewicz and Chung was published in 1967. • In the late 1960s and early 1970s, the FEM was applied to a wide variety of engineering problems. Origins of the Finite Element Method • The 1970s marked advances in mathematical treatments, including the development of new elements, and convergence studies. • Most commercial FEM software packages originated in the 1970s (ABAQUS, ADINA, ANSYS, MARK, PAFEC) and 1980s (FENRIS, LARSTRAN ‘80, SESAM ‘80.) • The FEM is one of the most important developments in computational methods to occur in the 20th century. In just a few decades, the method has evolved from one with applications in structural engineering to a widely utilized and richly varied computational approach for many scientific and technological areas

2014/5/15 Application of FEM in Engineering .MechanicalAerospace/Civil engineering/Automoble Available commercial FEM software packages engineering Thermal/Fluid flow NASTRAN (General purpose FEA) ectromagnetics PATRAN (Pre/Post processor) Process analysis (Injection molding.Forging.Casting.etc.) Hyper/Mesh (Pre/Post processor) Dyn-D (Impc/Crash analysis) 5

2014/5/15 5 Application of FEM in Engineering • Mechanical/Aerospace/Civil engineering/Automobile engineering • Structure analysis (static/dynamic, linear/nonlinear) • Thermal/Fluid flow • Electromagnetics • Process analysis (Injection molding, Forging, Casting, etc.) • … Available commercial FEM software packages • ANSYS (General purpose FEA) • ABAQUS (Nonlinear and dynamic analysis) • NASTRAN (General purpose FEA) • PATRAN (Pre/Post processor) • Hyper/Mesh (Pre/Post processor) • Dyna-3D (Impact/Crash analysis) • … Automotive System Analysis(combination of different element types)

2014/5/15 奈警被议全凭与实体单元的连按 shell elements and beam ement Contact problem analysis 轮与支撑轮之侧接敏的有限元分析 转轮的拓扑优化分析(Topology opm) tact analysis) 6

2014/5/15 6 大型转盘FEM分析 组合模型：板单元与实体单元的连接 大型转盘FEM分析- Large rotation disk FEM analysis 组合模型：板单元与梁单元之间的连接 Combination model: connections between shell elements and beam elements 大型转盘FEM分析：实体模型 Large rotation disk FEM analysis: Solid model Contact problem analysis 转轮与支撑轮之间接触的有限元分析 (Contact analysis) 转轮的拓扑优化分析 (Topology optimization)

2014/5/15 Heat transfor and thermal stross analysis of pistor 因 主态分 澀 - 的 7

2014/5/15 7 Heat transfer and thermal stress analysis of piston b) Aircraft System Analysis 破碎机主轴系统模态分析 (Mode analysis)

2014/5/15 Car(Plane)crash analysis 破碎锈和碎钢的碰撞力仿疯 Procsimulation-Forging Processsimulation-Stamping 装骤 Procssimulation-Casing Process simulation-Iniection molding 衡里且 1四 8

2014/5/15 8 Car (Plane) crash analysis 破碎锤和碎钢的碰撞力仿真 Process simulation - Forging Process simulation - Stamping Process simulation - Casting Process simulation – Injection molding

2014/5/15 New Developments in FEA Integrating FEA into CAD design sofware NOTE: Close relation with CAD Multi-scale analysis Multi-physics analysis 9

2014/5/15 9 NOTE: • FEM – The theoretical foundation for almost all CAE(Computer Aided Engineering) • Close relation with CAD