Chapter 2 Rudiment of Structural Analysis and FEM Chen Shenyan (陈珅艳) School of Astronautics, BUAA The New Main Building B1127 Tel:82338404 Email: chenshenyan@buaa.edu.cn

Review of last class

Structural optimization Definition: Combining the theories of mathematical programming and the methods of structural analysis to seek the best feasible design according to a pre-selected quantitative measure of effectiveness for loaded structures with computers as tools.

Mathematical Basis NLP Objective function constraints Design variables Feasible region

Structural optimization types 基础理论和技术 Structural optimization types Sizing optimization 尺寸优化 Shape optimization 形状优化 Topology optimization 拓扑优化 2019年2月18日

Sizing optimization 尺寸优化 10 杆桁架 10-bar truss 2019年2月18日

Optimization model Objective：minimum weight W Design variables：Cross-sectioanl areas Constraints： displacements, Stress, etc. 2019年2月18日

Shape optimization Original After optimization 2019年2月18日

Optimization model Objective：Minimum weight W Design variables：Coordinate lists Constraints： displacements stress or natural frequencies or buckling. 2019年2月18日

Topology optimization-truss structure Original design After optimization 2019年2月18日

Topology optimization of truss structure Objective：Minimum weight W Design variables：A Kept（ ）or remove（ ）； Constraints： displacements Stress or other responses as natural frequencies. 2019年2月18日

Topology optimization – continuum structure Design domain After optimization 2019年2月18日

Optimization model（Continuum structure） 引入虚拟的材料-刚度模型SIMP（ Solid Isotropic Microstructure/Material with Penalization for intermediate densities ） 2019年2月18日

Optimization model（Continuum structure） Minimum strain energy 2019年2月18日

Optimization model（Continuum structure） Minimum weight W 2019年2月18日

Mathematical Basis Kuhn-tucker conditions a regular point

Mathematical Basis Convex set Non-convex set

Mathematical Basis Convex-like function Non convex-like function

Mathematical model of typical structural optimization problem

Chapter 2 Rudiment of Structural Analysis and FEM

Content 2.1 General concepts in Structural analysis 2.2 Understand FEM through a simple example 2.3 General steps of FEM 2.4 Introduction of software Msc.Patran/Nastran

2.1 General concepts in structural analysis Refers to find the responses including stresses, displacements (deformations), vibration etc. in structures under loads or other environment conditions such as thermal gradients. Objective of Structural Analysis To answer questions as Does the structural work? Safe? As a Basis of Structural Design, to answer questions as “Is the structure needed to adjust? Where if necessary?”

2.1 General concepts in structural analysis （Cont） 3、 The ways or courses related to Structural Analysis Elasticity (elastic mechanics) and Plasticity (plastic mechanics)

2.1 General concepts in structural analysis （Cont） Finite difference scheme

2.1 General concepts in structural analysis （Cont） Material mechanics The course is mainly to study on deformation and stress in simple components of structures, such as bar, beam or simplified structural system constructed by them. Structural mechanics The course study on the responses for structures that consist of bar, beam and panel with special simplification. Generally the theories and methods to structural analysis in the course can be used to solve the problem with limited scale, types, efficiency.

2.1 General concepts in structural analysis （Cont） FEM (finite element method) It is a numerical analysis technique for obtaining approximate solutions to wide variety of engineering problems. It has been used on problems involving stress analysis, fluid mechanics, heat transfer, diffusion, vibrations, electrical and magnetic fields, etc.

Introduction to Finite Element Analysis The fundamental concept involves dividing the body under study into a finite number of pieces (subdomains) called elements (see Figure). Particular assumptions are then made on the variation of the unknown dependent variable(s) across each element using so-called interpolation or approximation functions. This approximated variation is quantified in terms of solution values at special element locations called nodes. Through this discretization process, the method sets up an algebraic system of equations for unknown nodal values which approximate the continuous solution. Because element size, shape and approximating scheme can be varied to suit the problem, the method can accurately simulate solutions to problems of complex geometry and loading and thus this technique has become a very useful and practical tool.

Finite Element Method (FEM) or Finite Element Analysis (FEA)

Finite Element Method Defined (cont.) 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.

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 (cont.) 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 (NASTRAN, 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.

2.2 Understand FEM through a simple example Truss structure

2.3 General Steps of FEM (1) Discretize the continuum (a) truss (b) Continuum

2.3 General Steps of FEM (cont) (2) Select interpolation function Interpolation function——to represent the variation of the field variable over the element. Selecting different interpolation function means selecting different element type for the model. Interpolation function or element type

2.3 General Steps of FEM (cont) (3) Find the element properties (element stiffness matrix) For the element load magnitude needed on the freedom degree of i if a unit displacement is raised on degree of freedom j and others remain fixed.

2.3 General Steps of FEM (cont) (4) Assemble the element properties to form the system stiffness matrix properties For the structural system load magnitude needed on the freedom degree of i if a unit displacement is raised on degree of freedom j and others remain fixed.

2.3 General Steps of FEM (cont) (5) Solve the system equation (6) Make additional commutations if desired, such as Stresses Reactions

2.4 Introduction of Software Msc.Patran/Nastran

Menu bar Application bar Tool bar 2019年2月18日 History list Command list

Governing Equation for Solid Mechanics Problems Basic equation for a static analysis is as follows: [K] {u} = {Fapp} + {Fth} + {Fpr} + {Fma} + {Fpl} + {Fcr} + {Fsw} + {Fld} [K] = total stiffness matrix {u} = nodal displacement {Fapp} = applied nodal force load vector {Fth} = applied element thermal load vector {Fpr} = applied element pressure load vector {Fma} = applied element body force vector {Fpl} = element plastic strain load vector {Fcr} = element creep strain load vector {Fsw} = element swelling strain load vector {Fld} = element large deflection load vector

Six Steps in the Finite Element Method Step 1 - Discretization: The problem domain is discretized into a collection of simple shapes, or elements. Step 2 - Develop Element Equations: Developed using the physics of the problem, and typically Galerkin’s Method or variational principles. Step 3 - Assembly: The element equations for each element in the FEM mesh are assembled into a set of global equations that model the properties of the entire system. Step 4 - Application of Boundary Conditions: Solution cannot be obtained unless boundary conditions are applied. They reflect the known values for certain primary unknowns. Imposing the boundary conditions modifies the global equations. Step 5 - Solve for Primary Unknowns: The modified global equations are solved for the primary unknowns at the nodes. Step 6 - Calculate Derived Variables: Calculated using the nodal values of the primary variables.