CHAPTER 1. Introduction

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1 ME 475: Computer-Aided Design of Structures 1-1 CHAPTER 1 Introduction 1.1 Analysis versus Design 1.2 Basic Steps in Analysis 1.3 What is the Finite Element Method? 1.4 Geometrical Representation, Discretization and Typical Finite Elements 1.5 Solution Validation

2 ME 475: Computer-Aided Design of Structures Analysis versus Design Analysis: Given loading conditions and the system properties, determine the performance of the system. In other words, given the inputs, determine the outputs. Example: Given the loading and cross-sectional geometry of an I-beam, determine its deflection. Linear analysis involves a direct solution procedure. Non-linear analysis typically involves an iterative solution procedure. In this class, we will use the finite element method to analyze the linear structural response of engineering systems. Design: Given loading conditions and the system performance targets, determine the system properties that satisfy those response targets. In other words, given the outputs, determine the inputs. Example: Given the loading and allowable deflection of an I-beam, determine the cross-sectional shape of the beam such that the mass is minimized and the deflection constraint is satisfied. Except in the simplest cases, design always involves an iterative process or search for a design that satisfies the performance objectives. In general, each iteration requires a complete analysis of a proposed design, so this process can be very computationally

3 ME 475: Computer-Aided Design of Structures 1-3 intensive. In this class we will use automated design optimization methods to identify the optimal design of engineering systems. Graphically, the design process can be described as follows: Initial Design Concept Specific Design Candidate Modify Design Build Analysis Model(s) Execute the Analyses No Design Requirements Met? Yes Final Design

4 ME 475: Computer-Aided Design of Structures Basic Steps in Analysis For any kind of engineering analysis: 1. Define the goals of the analysis. 2. Anticipate the physical behavior. 3. Identify a reasonable mathematical idealization (governing differential equations). a. Classify the predominant phenomena, and identify the most appropriate mathematical representation of each component. b. Taking into account the choice made in step 3a, determine an appropriate geometrical representation of the mathematical domain. c. Describe the boundary conditions. d. Identify key material behaviors (constitutive equations). 4. Solve for the desired response quantities (FEM, FDM, BEM, EFG, etc.). 5. Interpret and validate results and verify assumptions made in step Return to step 1, if necessary.

5 ME 475: Computer-Aided Design of Structures 1-5 These steps can be explained further as follows: 1. Define the goals of the analysis. Typical objectives are: Predict the stress or deflection at a particular point in a structure. Predict the first three natural frequencies of a structure. Determine the temperature distribution near a cutout. Predict the pressure variation in an internal flow. The goal(s) of the analysis will determine what kind of model is needed. This is a critical step, and one that should not be taken lightly. 2. Anticipate the physical behavior. For example: Stresses should be high near regions of geometric discontinuity. Thermal stresses should be small in regions that are loosely constrained. Turbulent flow often occurs near reentrant corners. Here, intuition is extremely valuable. It is sometimes helpful to perform a quick back of the envelope calculation to estimate certain response values.

6 ME 475: Computer-Aided Design of Structures Identify a reasonable mathematical idealization. a. Classify each part to be analyzed, and identify the most appropriate representation of each component. Possible representations include: Solid/Structural Mechanics: A continuum representation (based on elasticity theory): 1D a bar subjected to only axial load 2D a region exhibiting plane stress, plane strain, generalized plain strain, or axisymmetric response 3D a three dimensional solid A structural representation (based on mechanics of materials theory): Beam Plate/Shell A special representation: Spring Rigid link

7 ME 475: Computer-Aided Design of Structures 1-7 Fluid Mechanics: Ideal Inviscid (Potential) Flow Viscous Flow Non-Newtonian Flow Associated with each classification is a set of differential equations that govern the anticipated behavior and account for the assumptions made. Numerical methods provide approximations to these differential equations. b. Taking into account the choice made in step 3a, determine an appropriate geometrical representation of the mathematical domain. ** The mathematical and computational domain is that domain upon which the governing differential equations and boundary conditions are defined. It is not always necessary to explicitly represent every detail of a system. Often it is possible to represent the geometry in a simplified (or idealized) way without adversely affecting the predicted response in the

8 ME 475: Computer-Aided Design of Structures 1-8 primary region(s) of interest, provided the geometrical simplifications are not near the region(s) of interest. Small features, such as fillets and holes, may require significant effort to represent accurately. If these features do not affect the main results sought, as defined in step 1, then it may be possible to ignore them. This is sometimes called defeaturing the geometry. In the plate shown below, for example, the main region of concern may be the domain near the elliptical cutout. Since the rounded corners are a reasonable distance away from the region of primary concern, they may be represented as sharp corners. Likewise, the small holes near these corners may be neglected. This approach simplifies the geometry (and the analysis) but should have little or no effect on the predicted response near the cutout. Note that a clear definition of analysis objectives (step 1) is required to justify such simplifications. Actual Geometry Idealized Geometry

9 ME 475: Computer-Aided Design of Structures 1-9 c. Describe boundary conditions. These should be represented as accurately as possible. However, if the primary region(s) of interest are a significant distance away from the location of loading or constraint, it may be possible to simplify the representation of these boundary conditions (e.g., using Saint Venant s Principle). e. Identify key material behaviors. For many classes of problems, especially nonlinear ones, this aspect of the model is often the most challenging and the most important. 4. Solve for the desired response quantities. For example: Solid Mechanics: Displacements Strains Stresses Natural frequencies Buckling loads Transient response

10 ME 475: Computer-Aided Design of Structures 1-10 Fluid Mechanics Velocity Pressure Temperature 5. Interpret and validate the results and verify the assumptions made in step Return to step 1, if necessary. When defining a problem, there are two extreme approaches. 1. Simplify the mathematical representation to a form that allows a simple analytical solution. In this case, important physical behaviors may be neglected, leading to erroneous predictions. This approach is often useful in the early stages of an analysis or design effort, and is always useful to validate more detailed analyses. 2. Represent all existing physical behaviors in the mathematical model and obtain a computational solution. In this case, mathematical approximations and/or algorithm shortcomings may yield errors in predictions. The optimal approach often lies somewhere between these two bounds, and can usually be identified based on good engineering judgement and experience.

11 ME 475: Computer-Aided Design of Structures 1-11 When used correctly, present day computational methods and software can often produce solutions that are at least as accurate as the input data (material properties, loads, constraints, etc.). In other words, there is often at least as much error in the specification of the problem data as there is in the numerical solution of the differential equations. Nevertheless, only those features and phenomena deemed important should be included in a given model.

12 ME 475: Computer-Aided Design of Structures What is the Finite Element Method? Basic Steps: 1. Break up (discretize) the mathematical domain into many simple domains. element node A mesh of 3-noded triangular elements 2. Develop element equations that approximately describe the phenomena occurring within each element. The element equations are based on spatial approximations of the field variables (e.g., displacements or temperatures) within each element. These approximation functions are almost always polynomials. The resulting element equations are algebraic equations.

13 ME 475: Computer-Aided Design of Structures Enforce continuity of the solution across element boundaries, apply boundary conditions, and solve. 4. Check the accuracy of the solution. Both validation and convergence should be evaluated. These will be discussed in greater detail later in the course. Most physical phenomena can be well described by differential equations. These equations can be very difficult to solve, especially for problems with complex domains, loading or material properties. ** The essence of the finite element method (FEM) is to approximate the solution to differential equations by replacing them with a set of easily solvable algebraic equations based on subdomain (element) approximations. Thus, a differential equation that is difficult to solve is converted into a set of algebraic equations that are easily solved. The equations for each element are coupled to those equations in adjacent elements by enforcing continuity of the primary DOFs at the nodes. Thus a global system of equations is obtained that describes the solution in the entire domain. These equations relate generalized nodal displacements {U} to generalized nodal forces {F} through the equations [K]{U}={F}, where [K] is called the stiffness matrix.

14 ME 475: Computer-Aided Design of Structures Geometrical Representation, Discretization and Typical Finite Elements Physical systems and structures are three-dimensional. However, for a given problem classification, the associated governing differential equations may be onedimensional, two-dimensional or three-dimensional. The particular form and dimensionality of the governing differential equations depends upon the assumptions about how the system or structure will behave. ** Since the finite element method provides approximate solutions to these differential equations, the dimensionality of the computational domain will be the same as that of the mathematical domain over which the differential equations are defined. Example: In an axisymmetric problem, the geometry, material properties and boundary conditions do not vary in the circumferential direction. Hence, the solution of interest does not vary in the circumferential direction (as long as the problem is linear). So the three-dimensional problem can be reduced to two dimensions in the r-z plane. The corresponding differential equations are two-dimensional, as is the computational domain.

15 ME 475: Computer-Aided Design of Structures 1-15 θ r z a. Three-dimensional physical domain of a cylinder. θ r z b. Two-dimensional computational domain of the axisymmetric problem.

16 ME 475: Computer-Aided Design of Structures 1-16 Example: A beam with a complex cross-section may be subjected to a threedimensional loading state. Nevertheless, the overall beam behavior is usually predicted well using beam theory, which is described using onedimensional (ordinary) differential equations. So the three-dimensional problem can be reduced to a one-dimensional problem. The corresponding differential equations are one-dimensional, as is the computational domain. y x z a. Three-dimensional physical domain of the beam. x b. One-dimensional computational domain.

17 ME 475: Computer-Aided Design of Structures One-dimensional elements two-noded linear element three-noded quadratic element Two-dimensional elements three-noded linear triangle six-noded quadratic triangle four-noded bilinear quadrilateral eight-noded biquadratic quadrilateral

18 ME 475: Computer-Aided Design of Structures Three-dimensional elements four-noded linear tetrahedral ten-noded quadratic tetrahedral six-noded linear wedge fifteen-noded quadratic wedge eight-noded tri-linear hexahedral (brick) twenty-noded tri-quadratic hexahedral

19 ME 475: Computer-Aided Design of Structures Solution Validation Effective and responsible use of computational methods, such as the finite element method, requires a thorough understanding of the physical nature of the problem being solved, the mathematical description of the problem, and the nature or behavior of the computational algorithms and approximations. Without this knowledge, one often cannot create a suitable model or evaluate the quality of the results. Finite element results must always be validated against some other form of solution: analytical experimental computational

20 ME 475: Computer-Aided Design of Structures 1-20 The solution used for validation should be independent of the finite element solution being validated. For example, a re-solution of the problem using a modified or refined mesh is good practice, but does not constitute a proper solution for validation. If a finite element solution is used to validate another finite element solution, then the two models should be created independently, and the solutions should ideally be obtained using different codes, in case one of them is erroneous.

1.2 Numerical Solutions of Flow Problems

1.2 Numerical Solutions of Flow Problems DIFFERENTIAL EQUATIONS OF MOTION FOR A SIMPLIFIED FLOW PROBLEM Continuity equation for incompressible flow: 0 Momentum (Navier-Stokes) equations for a Newtonian