FEA Final Exam

FEA Final Exam A partial differential equation is the mathematical description of a ____________ physical process in which a dependent variable, u, is a function of _____________ independent variables, e.g, t, and x. ...

FEA Final Exam A partial differential equation is the mathematical description of a ____________ physical process in which a dependent variable, u, is a function of _____________ independent variables, e.g, t, and x. In order to solve a PDE we need to know a set of __________. - ✔✔Continuous, one or more, boundaries Assuming the principles of a standard discrete system: What is the dimension of an element stiffness matrix? - ✔✔m X m m is the number of nodes in the element Assuming the principles of a standard discrete system: What is the dimension of the global stiffness matrix after assembly of the whole structure? - ✔✔a X a a is the number of global nodes in the element and the summation of the local element Basic Workflow of FEA Model ______ Collect ______ Build _______ _______ to FE ______ Assign ______ Apply _______ Run analysis _____ of results _____ with calibration Analysis of results - ✔✔Sketch Data Model(geometry) Discretize, mesh Materials and Properties BCs Sanity Check Iterate Complete the formula u=summation __*__ - ✔✔Na=shape function Ua=approximate displacement vector Considering that we are given the partial differential equation for a physical problem. Why is it necessary for some cases to use numerical methods such as the finite element method? - ✔✔Sometimes we can't get the exact solution of the known so we use the finite element method to approximate the solution. explain .dat file - ✔✔data file- contains comments on job run and used to check errors explain .hm file - ✔✔hypermesh file- where the model is created and meshed explain .inp file - ✔✔input file- text file where material properties and BCs are inserted explain .odb file - ✔✔output database results file- used in hyperview to view stresses and displacements Explain strong form - ✔✔the strong form of the problem is described by a differential equation and boundary conditions. -d^2phi/dx^2 +Q(x)=0 Explain week form - ✔✔the weak form of the problem is an integral function multiplied by an arbitray v and is set equal to zero. ∫?[−?2???2+?(?)]??=0 Galerkin method is the method of _________________. - ✔✔Weighted Residuals Galerkin's method gives an approximate solution for our problem. What is the choice of our weighting function in Galerkin's method? - ✔✔Wb=Nb How do we find a solution for the complete domain? (Write down the new equation and explain what you did). - ✔✔???+??=0 We set the equation equal to zero to balance out the forces on the nodes. (equilibrium). For example, for node a, Σ???=??1+??2+⋯+???=0. Then ???+??=0. The sum of all the nodal forces contribute to node a is equal to zero. In difference to a standard discrete system the more general finite element method utilizes an ____________ of our unknown parameters. - ✔✔Approximation In order to run an analysis which keywords have to be listed between *STEP and *END STEP (assume you will use gravity)? - ✔✔*Static *DLOAD *Boundary In the following *ELEMENT command, what do the various numbers stand for, and what type of element is given here? *ELEMENT, Type CPE4, ELSET=NAME 123,456,789,2345,6789 - ✔✔Element #, node #'s that make up the element Continuous Plain Strain (4 nodes, quad) List at least 2 possibilities where a rapid change in stress or strain is happening. - ✔✔Around sharp corners Around pointed edges Fault/Near wellbore Principal Stresses - ✔✔In a particular coordinate system, 3 mutually perpendicular normal stresses where all shear stresses go to zero. Reasons meshing could be poor quality? - ✔✔Distortion No Biasing Finer mesh should be near well bore T or F A one dimensional domain of 11cm can be discretized into 23 elements. - ✔✔True T or F A principal state of stress is given by the following stress tensor: ?=(12 −1 −1 12) - ✔✔False T or F Element numbering in a discretization affects the band-with of the global stiffness matrix. - ✔✔False, nodal T or F Essential boundary conditions are prescribed values of the first order derivative of the unknown on the domain boundary. - ✔✔False T or F For a second order PDE we need 2 BCs. - ✔✔True T or F Higher order elements give more accurate results. - ✔✔True T or F Na (xb, yb)=1 - ✔✔False T or F Na (xb,yb)=0 - ✔✔True T or F Principal stresses represent the highest shear stresses. - ✔✔False T or F Principle stresses are ALWAYS perpendicular to each other. - ✔✔True T or F Shap functions with higher order approximation functions give more accurate results. - ✔✔True T or F Shape functions help us to determine the solution of the unknown parameter at any point within an element. - ✔✔True T or F Shape functions help us to determine the solution of the unknown parameter at any point within the element. - ✔✔True T or F The basic idea of Galerkin's approximation is to seek an approximate solution of a weak form for the element domain rather than for the whole domain. - ✔✔True T or F The basic idea of Galerkin's approximation is to seek an approximation solution of a weak form for the element domain rather than for the whole domain. - ✔✔True T or F The Euler equations derived from a functional are equal to the strong form of the problem. - ✔✔True T or F The finite element method approximates the partial derivatives by utilizing difference equations. - ✔✔False, equilibrium equations T or F The following state of stress is not possible: ?=(15 −1.5 −1.5 15) - ✔✔False T or F The node numbering in a discretization does not matter. - ✔✔False T or F The state of stress is given in its entirety by three mutually perpendicular normal stresses. - ✔✔False, only in coordinate system where shear stresses vanish T or F Three mutually perpendicular normal stresses are defined as principal stresses. - ✔✔False T or F Variational Principles are equal to the minimization of the potential energy. - ✔✔True The *NODE command looks like.... - ✔✔*NODE 1,5.5,2.3,6.7 -the numbers stand for the node number and then the coordinates of the nodes The finite difference method utilizes approximations to find a solution to a PDE. What exactly is approximated? - ✔✔The answers are approximated using subcontinuums, each defined as a finite element. The changes in each element when it moves and the forces acting on that element are what is approximated. The finite element method seeks a solution in an __________ form. The finite element method seeks a solution in an _________ form. Give the general formula for this and define the parameters. - ✔✔Approximation (fill in the blank) ?=Σ????? ??: Shape function ??: Unknown approximate displacement vector The state of stress at any given point is given by_______. - ✔✔Stress Tensor The traction vector on any surface is calculated by __________. - ✔✔??=????? The virtual work principle for a linear elastic plane strain problem results in equations in _______ form and the resulting equation to solve is _______. - ✔✔integral, ku+f=0 To obtain the weak form of the problem we sometimes have to integrate by parts. When or why is this procedure necessary? - ✔✔We integrate by parts to get rid of the higher order derivative. This needs to be done so we can apply the BC's and to avoid continuity. Using the *BOUNDARY keyword for a nodeset "Bottom" in a 2D model, how do you set up a "Pin" and "Roller"? - ✔✔*Boundary NSET=Bottom, DOF(from), DOF(to), displacement Roller - Bottom,2,2,0.0 Pin - Bottom,1,2,0.0 We use fine mesh in areas where we expect what? - ✔✔Rapid change in stress or strain What are essential boundary conditions? - ✔✔Essential boundary conditions are prescribed values for the unknowns in the domain. What are the 3 general sections of the input file. - ✔✔Material Solid Section Load Section Geometry What does biasing mean in terms of FE discretization? - ✔✔fine mesh near zone of interest and coarser mesh away from it What does CPS4 mean? - ✔✔Continuum plan stress (quad elements) What does q=ku+f=0 stand for? - ✔✔K=stiffness matrix u=displacement (unknown) f=element force vector q=nodal force e=element number *need to know boundary conditions What does the keyword *SOLID SECTION, ELSET=....., MATERIAL=..... do? - ✔✔Assigns material properties to the elements in the ELSET What information should be given in a model sketch? - ✔✔Geometry Material Boundary Conditions Load Types Dimensions of Geometry What is the difference between a standard discrete system and the finite element method? - ✔✔Finite element method is an approximate solution in integral form which allows arbitrary discretization. Standard discrete system is a predefined discretization with exact solution at nodes. What is the meaning of discretization? - ✔✔When the continuum is broken into regions known as finite elements What is the most general form of the equation of motion? - ✔✔F=ma What is the name of the meshing software we use? - ✔✔HyperMesh What is the structure of the input file? give 3 major sections - ✔✔Geometry Materials Loads What other possibilities do we have to obtain ku+f=0 from an approximate solution? - ✔✔Raleigh Ritz Method What's the requirement for our weighting functions if we have essential boundary conditions? - ✔✔The requirement for the weighting functions is that they go to zero if we have EBC. Where do we use a fine mesh? - ✔✔A fine mesh is used in areas of rapid change in stress and strain. Why or when do we need to use numerical models? - ✔✔Simulation of physical processes requiring a complex set of BC's where analytical solutions do not exist. You calculate the stiffness matrix of a displacement based element and you find it to be: - ✔✔?(1 -0.513 ) (−0.513 1). Do you trust your result? Give a reason. Yes, because it gives a symmetric stiffness matrix, but we would have to double check it.