1. Chapter 2 Rudiment of Structural Analysis and FEM
Chen Shenyan (陈珅艳)
School of Astronautics, BUAA
The New Main Building B1127
Tel:

2. Review of last class

3. 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.

4. Mathematical Basis NLP Objective function constraints Design variables
Feasible region

5. 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

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

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

8. Shape optimization
Original
After optimization
2019年2月18日

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

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

11. 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

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

13. 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

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

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

16. Mathematical Basis
Kuhn-tucker conditions
a regular point

17. Mathematical Basis
Convex set
Non-convex set

18. Mathematical Basis
Convex-like function
Non convex-like function

19. Mathematical model of typical structural optimization problem

20. Chapter 2 Rudiment of Structural Analysis and FEM

21. 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

22. 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?”

23. 2.1 General concepts in structural analysis （Cont）
3、 The ways or courses related to Structural Analysis
Elasticity (elastic mechanics) and Plasticity (plastic mechanics)

24. 2.1 General concepts in structural analysis （Cont）
Finite difference scheme

25. 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.

26. 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.

27. 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.

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

29. 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.

30. 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 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.

31. 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.

32. 2.2 Understand FEM through a simple example
Truss structure

33. 2.3 General Steps of FEM (1) Discretize the continuum
(a) truss (b) Continuum

34. 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

35. 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.

36. 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.

37. 2.3 General Steps of FEM (cont)
(5) Solve the system equation
(6) Make additional commutations if desired, such as
Stresses
Reactions

38. 2.4 Introduction of Software Msc.Patran/Nastran

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

40. 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

41. 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.