INTRODUCCION AL ANALISIS DE ELEMENTO FINITO (CAE / FEA)

Transcription

1 INTRODUCCION AL ANALISIS DE ELEMENTO FINITO (CAE / FEA)

2 Title 3 Column (full page) 2 Column What is Finite Element Analysis? 1 Column Half page

3 The Finite Element Method The Finite Element Method (FEM) is a computational technique developed by engineers used to obtain approximate solutions to engineering problems.

4 How the FEM works - Divide and Conquer! Partial differential equations describe nature (continuous) In the FEM, a continuous domain is discretized using functions which give us a very good approximate solution With the advent of computers, very large linear systems of equations can be solved. Partial Differential Equations (PDEs) System of Linear Equations (Ax=b)

5 How the FEM works - Divide and Conquer! In the FEM, a continuous domain is discretized into simple geometric shapes called elements. A Finite Element is a small piece of a structure Nodes appear on the element boundaries and fasten the elements together Continuum : infinite number of degrees-of-freedom (DOF) Discretized model : finite number of DOF

6 How the FEM works - Divide and Conquer! Step 1: Discretize the continuum Step 2: Select interpolation functions Step 3: Find the element properties Step 4: Assemble the element equations Step 5: Solve the global equation system Step 6: Compute additional results (stress, strain, etc.)

7 Example: 1D axially loaded bar Structure Finite Element Representation

8 Example: 1D axially loaded bar Consider the prismatic elastic bar with: Length of (L) Elastic modulus (E) Cross-sectional area (A): Bar/Truss element u1 F1 u2 Axial loading and deformation F2

9 Deriving the stiffness matrix [K] u1 u2 Constitutive equations: F2 F1 (1) (2) Watch this derivation at:

10 Deriving the stiffness matrix [K] (1) (2)

11 Governing equation where: [K] = stiffness matrix {u} = nodal displacement vector {F} = Force/load vector

12 Global assembly of equations u1 F1 u2 Local equations F2 Global equations

13 Essential boundary condition (Dirichlet) Refers to a fixed boundary condition This is equivalent to setting u1= 0 in the global system of equations

14 Natural boundary condition (Neumann) Refers to the Force boundary conditions that vary over time.

15 Solve the global equation system Solve for the unknown nodal displacements

16 Compute additional results (stress, strain, etc.) The strain within an element: We use special shape functions (N) to interpolate the solution between the discrete values obtained at the mesh nodes The stress within an element:

17 FEM in 1D, 2D, and 3D 1D, 2D, and 3D FEM follow the same general principles Different element types

18 1-Dimensional Elements Connect 2 nodes in a straight line The cross-sectional properties are defined by the finite element properties Applicable for analyses where the members are long, slender and have a constant cross-section

19 2-Dimensional Elements Structural element with one small dimension and 2 larger dimensions Rule of thumb: the small dimension should be at least 1/15 the size of the larger dimensions Flat elements (plates)

20 3-Dimensional Elements Contain all of the geometric definition in the element geometry Volume elements

21 Errors in FEA Modeling Error We are not analyzing the physical model, but a simplified mathematical model Discretization Error Errors that arise from the creation of the mesh Infinite domain vs. finite domain Numerical Error Introduced by the computer when it rounds or truncates numbers as it assembles matrices and solved equations

22 Linear vs. Nonlinear Linear Nonlinear The relationship between force and displacement is linear The relationship between force and displacement is nonlinear Material behavior remains in the elastic zone Material yields (plastic zone) or other material non-linearity Only valid for small deflections and rotations Large deflections Examples: moving parts, non-metallic materials such as composites or rubber, impacts loads

23 Static vs. Dynamic Static Dynamic Static refers to a model which is loaded slowly, thus inertial forces and damping are ignored Dynamic loads are applied as a function of time or frequency Single time step - not a function of time Examples are shock, vibration, and seismic loading where the load amplitudes vary significantly with respect to time More complicated and more realistic than static analysis

24 Responsibility of the user Smooth and colorful stress contours can be produced by any model, good and bad A responsible user must understand the nature of the problem and the inherent assumptions before setting up the problem and analyzing the results

25 In summary FEA is a crucial step in the product design process that allows you to test your design for stress, vibration, buckling, and fatigue BEFORE prototyping Engineering calculations Physical testing

26 Title 3 Column (full page) 2 Column Stress analysis of a reclaimer arm 1 Column Half page

27 Stress analysis of a reclaimer arm

28 Stress analysis of a reclaimer arm Boundaries and Loading Conditions fx Force Load fx = N fy = 0N fz = N Fixed faces at base dx = 0 dy = 0 dz = 0 fz Z Y X

29 Link to Project

30 Title 3 Column (full page) Column What is 2Computational Fluid Dynamics (CFD)? 1 Column Half page

31 Computational Fluid Dynamics Computational Fluid Dynamics (CFD) is a field in fluid dynamics that incorporates numerical analysis to simulate and solve problem involving fluid flows. Basically can be regarded as numerical experiments. The numerical calculations are performed on computing machines (computers) to solve conservation equations for fluid dynamics. This provides fast and cost-effective insight of the flow problem for better performance and design of the process or product

32 Physical Problem to Simulation Model Types of Flows Quantity of interest Incompressible / Compressible Density, ρ Steady-state / Transient Time, t Laminar / Turbulent Reynolds number, Re or Velocity, v Inviscid / Viscous Viscosity μ Sub- / Transonic Mach number, M Single- / Multi-Phase Macroscopic fluid properties

33 Physical Problem to Simulation Model Navier-Stokes equations: Mass conservation Change of mass in time Flow of mass through the boundaries Mass does not disappear or appear from nowhere

34 Physical Problem to Simulation Model Navier-Stokes equations: Momentum conservation Time change Convection term Similar to mass transport. This time we move momentum Pressure forces Viscous force Extra forces (gravity, etc.) Forces that act on the fluid. They generate / dissipate momentum.

35 The Finite Volume Method Most equations are derived by considering a very small fluid volume called Control Volume (CV) and applying the conservation laws Since each CV has finite size, this method is called Finite Volume Method ( FVM) The entire continuous flow domain is discretized into these simple shaped Control Volumes (CVs).

36 The Finite Volume Method The Control Volume: Face centroid CV centroid CV boundary

37 The Finite Volume Method The Control Volume: All fields are stored in the centroid. Data between them is interpolated. Velocity (U) vector field Pressure (p) scalar field Temperature (T) scalar field

38 How CFD works Step 0: Physical problem to Mathematical model Step 1: Meshing generation (discretize the continuum), Pre-processing Step 2: Select Type of analysis (simplified model version of problem) Step 3: Specify the Fluid properties or model Step 4: Specify Boundary conditions Step 5: Select Solvers and Interpolation Schemes for solution of equation system Step 6: Post-processing results e.g Pressure loss/gain, Vorticity, Forces

39 How CFD works - Process

40 Errors in CFD Modeling and Setup Error: 1st biggest source of errors A simplified mathematical model, replicating physical setup & conditions Discretization Error: 2nd biggest source of errors Errors that arise from a bad or inadequate mesh Domain approximations Numerical Error Introduced by the computer when it rounds or truncates numbers as it assembles matrices and solved equations

41 Errors in CFD Modeling Errors: Outlet Sections

42 Errors in CFD Modeling Errors: Inlet Sections Inlet Outlet Flow area of interest

43 Errors in CFD Modeling Errors: Inlet Sections Inlet too close to area of interest Inlet at proper distance to area of interest

44 Errors in CFD Modeling Errors: Taking physics into account Fluid entering container Outlet for Air in the container

45 Responsibility of the user Smooth and colorful contours can be produced by any model, good or bad A responsible user must understand the nature of the problem and the inherent assumptions before setting up the problem and analyzing the results The results should always be first compared qualitatively and then with Analytical calculations or Hand approximations to check for obvious inconsistencies

46 CFD in SimScale 1 CAD Import 2 Mesh & Simulate 3 Design Decision

47 Are my results correct?

48 Are my results correct? Following criteria can also help to judge if a simulation is converged, which is a necessary but not sufficient criterion Residuals Forces Probes

49 Title 3 Column (full page) 2 Column Incompressible Analysis of Laminar Pipe Flow 1 Column Half page

50 Laminar Steady Pipe Flow Fixed Flow Inlets Wall Flow Outlet

51 Sample Case: Pipe Flow problem Problem Overview: Velocity: Low Flow behaviour: Independent of time Fluid: Water Fluid: Water Velocity: Low Interested in time independent flow Incompressible Analysis Laminar Flow Steady-State

52 Project Link Tutorial reference: Exercise reference:

53 Title 3 Column (full page) 2 Column Overview of Thermal Simulations 1 Column Half page

54 Overview of Thermal Simulations What is Thermal Simulation? Thermal simulations can be used to calculate the temperature and heat transfer between different components and their environment The properties on which thermal simulation The products must be investigated for temperature, heat dissipation/removal, thermal stresses, flow field of fluids, density changes and so on. Importance of thermal simulation At any stage of design, thermal simulation can help designers and engineers to ensure that each component of an assembly performs properly within expected temperature ranges

55 Title 3 Column (full page) 2 Column Thermal simulation on structures Introduction to Thermal Simulation 1 Column Half page

56 What is Finite Element Analysis? The Finite Element Method (FEM) is a computational technique developed by engineers used to obtain approximate solutions to engineering problems.

57 Title 3 Column (full page) 2 Column Thermal simulation on structures Three different heat transfer phenomena 1 Column Half page

58 Three different heat transfer phenomenon Conduction What is conduction? Transfer of heat between substances that are in direct contact with each other Exists in all phases Heat energy passes from hot to the cold end of the substance Better the conductor - more rapid heat transfer e.g. metals are good conductor of heat compared to plastics Simulation examples Conduction of heat in different material (see image on the right) Heat transfer in a spark plug Heat transfer in sink through a LED mounted on it Heat transfer via conduction in steel, ABS plastic, copper and lead

59 Three different heat transfer phenomenon Convection What is convection? Transfer of heat from substance by movement of particles In fluids - liquids and gases Surface phenomenon Convection depends on the velocity of liquid or gas i.e. the faster moving liquid or gas will take out more heat from substance than that of the static one. Simulation examples Data center cooling (see image on the right) Convection inside light bulb Cooling of a heat sink Cooling of a data center via convective heat transfer

60 Three different heat transfer phenomenon Radiation What is radiation? Transfer of energy through waves No medium required Heat can be transferred even in empty space by thermal radiation Radiation predominant at higher temperatures Simulation examples Immense heating of a Spotlight (see image on the right) Cooling of highly heated steel plate Effect of radiative heat transfer on spotlight heating

61 Design Process Step 1: Create geometric model Model with CAD software

62 Design Process Step 2: Discretize flow domain Create a mesh of the fluid domain

63 Design Process Step 3: Define the physics model All other walls defined with standard room temperature Refrigerator coils defined with constant temperature Apply Boundary Conditions

64 Design Process Step 4: Results evaluation Visualize Temperature, Velocity etc.

65 Books & resources [Book] Introduction to the Finite Element Method Niels Ottosen & Hans Petersson [Book] The Finite Element Method Set (Sixth Edition) O.C. Zienkiewicz, R.L. Taylor and J.Z. Zhu [Book] A First Course in Finite Elements Jacob Fish, Ted Belytschko [Book] Concepts and Applications of Finite Element Analysis Robert D. Cook and David S. Malkus [Book] Finite Element Procedures K.J. Bathe (MIT) [Video Lectures] Finite Element Procedures for Solids and Structures K.J. Bathe (MIT) [PDF] Introduction to Finite Element Methods University of Colorado at Boulder [PDF] Introduction to the Finite Element Method - G. P. Nikishkov

66 Books & resources An Introduction to Fluid Dynamics by G. K. Batchelor Physical Fluid Dynamics by D. J. Tritton Fundamentals of Aerodynamics Book by John D. Anderson Modern Compressible Flow: With Historical Perspective 2nd Edition Book by John D. Anderson Modern Compressible Flow: With Historical Perspective by J. D. Anderson Elements of Gasdynamics by H. A. Liepmann and A. Roshko. Liepmann and Roshko Boundary Layer Theory by H. Schlichting Turbulent Flows by Stephen B. Pope Thermodynamics, Schaum s Outline Series by M. M. Abbott and H. A. van Ness