DUE: WEDS MARCH 26TH 2018

Transcription

1 HOMEWORK # 2: FINITE DIFFERENCES MAPPING AND TWO DIMENSIONAL PROBLEMS DUE: WEDS MARCH 26TH 2018 NOTE: In this homework, you will choose ONE of the following three questions to perform and hand-in. Each question has a bonus component. In the introduction of each question, the mathematical and coding difficulty is estimated. You may use this to determine which problem to solve. All questions have the same points allotment and the same amount of bonus points available. 1. Question # 1: Beam Bending Notes: This question is mathematically quite intense (most intense mathematical problem); however, it is less Matlab/computationally challenging as others since you already have worked on a code for beam bending for uniform node distributions. The mathematical challenge is due to deriving the 4th-order PDE mapping equation for the problem. You are encouraged to watch and work through the mapping Video 11 prior to starting this question ( In this question we revisit the beam bending problem to examine the impact of mesh/discretization refinement on the accuracy of the solution. The problem considers a constant cross section (I = const.) and constant material properties (E = const.) beam with a non-uniform mesh (varying dx) distribution along the length of the beam. The ultimate idea is to see whether clustering nodes at different locations along the beam can help to reduce the error when the same number of grid points are used in a beam bending calculation. The governing equation in the physical domain is the fourth order beam partial differential equation: (1) (EI) 4 w x 4 = q(x) With the following boundary conditions in the physical domain: (2) w w (x = 0) = 0; x (x = L) = 0; w(x = 0) = 0; w(x = L) = 0 x 1

2 2 DUE: WEDS MARCH 26TH 2018 Figure 1. The beam physical domain, and boundary conditions. In all but the bonus question, the following expression will be used to map from the computational space, ξ = [0, 1], to the physical space x: (3) x(ξ) = ( ( )) cos π 3ξ cos( π 8 ) cos( 7π 8 ) cos( π 8 ) The following properties will be assumed: The beam will have a constant cross section with I = m 4 The beam will have a length, L = 1m The beam is aluminum, E = Pa q(x) = 55 N/m. Please answer the following questions (hand-in a paper copy of answers, code will be uploaded via turnitin). (1) On separate graphs, plot the beam nodes in both the physical and computational spaces. (2) Calculate the first, second, third and fourth derivative of the mapping function with respect to the computational variable ξ, i.e., dx dξ, d2 x, d3 x dξ 2 dξ 3 and d4 x dξ 4. (3) Download and run the 1-D string mapping equation from the course website and modify this code to solve the string problem using the mapping expression for this question. Verify your implementation of this change is correct. (4) Derive the beam bending equation that will be used/solved in the computational domain for an arbitrary distribution of nodes in the physical domain. Please factor the final answer so that the equation has the form: A 4 w ξ 4 +B 3 w ξ 3 +C 2 w ξ 2 +D w ξ = q(ξ). NOTE: This question/equation derivation is the bulk of the effort for this question (and the equation is pretty messy) so, please take care.

3 HOMEWORK # 2: FINITE DIFFERENCES MAPPING AND TWO DIMENSIONAL PROBLEMS 3 (5) Please provide both the continuous and discrete representation for the boundary condition in computational domain (i.e. map the relevant boundary conditions to the computational domain). (6) Write the psuedo code for your Matlab solver clearly show how and where the domain mapping components will be incorporated into the code. (7) Write/modify (from HW#1) your beam bending computer code to solve this mapped problem (in Matlab or other programming language). Remember that the beam has constant cross section and constant material properties. (8) Show that your mapping version of the code matches theoretically calculated deflections (you can find the theoretical solution to this problem online or in a solid mechanics textbook) and calculate and plot the error convergence (clearly state what norm you are using and how you are computing the error). If you find that you are not getting the right answer and you suspect your mapping, you can try simpler polynomial mappings to isolate any potential bugs such as: x(ξ) = (ξ+0.5) , or x(ξ) = (ξ+0.5) , or x(ξ) = (ξ+0.5) (9) BONUS: Implement finite difference expressions to calculate the mapping derivatives/parameters and determine the best/optimal distribution of nodes in the physical space to minimize the beam bending max deflection error. Use 41 nodes to represent the beam.

4 4 DUE: WEDS MARCH 26TH Question # 2: Non-Rectangular Domain Membrane Deflection Notes: This question is moderate Matlab and moderate math difficulty. You are encouraged to watch the 2D finite difference mapping theory videos (Video 12 & 13) prior to starting this question. Note you may also want to download and use the code on the website as a starting point (note: this code is not perfect, so you ll need to understand it). In this question you will solve the transverse deflection of an annular and elliptical membrane due to uniform pressure loading. In the physical space, the governing differential equation is: (4) 2 w = 2 w x w = f(x, y) = 1 y2 The geometry is defined using the following equations: MajorAxis(ξ) = (MajorAxisMax MajorAxisMin)ξ + MajorAxisMin MinorAxis(ξ) = (MinorAxisMax MinorAxisMin)ξ + MinorAxisMin x(ξ, η) = M ajoraxis(ξ) cos(2πη) y(ξ, η) = M inoraxis(ξ) sin(2πη) The following diagram shows the geometry in the physical vs. computational spaces: Figure 2. The membrane mapping problem the elliptical bundt cake pan.

5 HOMEWORK # 2: FINITE DIFFERENCES MAPPING AND TWO DIMENSIONAL PROBLEMS 5 The boundary conditions are shown in figure 3. Dirichlet boundary conditions (deflection = 0, which can be mathematically expressed as: w(x, y) = 0) are applied to the outer boundaries of the domain. The top and bottom boundaries are such that continuity of the solution is applied there (a periodic boundary condition, w upper = w lower ). Figure 3. The membrane boundary conditions (physical and computational domains). Please answer the following questions (hand-in a paper copy of answers, code will be uploaded via turnitin). (1) Setup the computational domain in Matlab such that ξ = [0, 1] and η = [0, 1]. Show that when the mapping equations are applied that the computational domain maps to the correct physical domain (i.e., plot the mesh/vertices in the computational and physical spaces). (2) Determine the relevant expressions for the first and second derivatives of the mapping equations with respect to the computational domain variables. E.g.: dx dξ, d 2 x dξdη., etc. (3) Write the partial differential equation that will be solved in the computational domain to solve the physical membrane deflection problem. (4) Clearly present the relevant finite difference expressions (for the unknown membrane deflection terms) that will be used as well as draw an illustration of the nodal computational stencil for a point i, j inside the domain. (5) Clearly show the entries that will be inserted into row i of the matrix for a difference equation centered at node, i, j, inside the domain (not a boundary node). (6) Starting with the polar coordinates poisson equation, 2 w = 2 w + 1 w r 2 r r w = r 2 θ 2 1, derive/determine the analytical solution of the membrane deflection for a circular annular physical space with an inner radius, R inner = 1 and an outer radius,

6 6 DUE: WEDS MARCH 26TH 2018 R outer = 4. Plot the deflection as a function of the radial location in the annular section. Note: since the problem is axially symmetric, any dn dθ terms are zero. n (7) Write a psuedo code for solving the membrane problem in computational space for different annular domains defined by inner and outer ellipses. (8) Write your matlab code and check that it works. (9) Plot your error convergence based on the comparison of your computed solution to the analytical solution you derived. Clearly describe the calculation of the error so that it is clear what is being shown in the convergence plot. (10) Experiment: Assuming that the inner boundary of the domain is a unit radius circle (MajorAxisMin = 1 and MinorAxisMin = 1), and the outer major and minor radii vary, surf plot the maximum deflection of the membrane for different outer boundary major and minor axis radii (MajorAxisMax = [1.2, 1.4,..., 6] and MinorAxisMax = [1.2, 1.4,..., 6]). Use a mesh that has 41 nodes in the ξ and η directions. (11) Bonus: Rewrite the mapping portion of the code so that it uses finite difference expressions based on the computational vs. physical meshes to calculate the mapping. Run the same computations as the prior experiment with numerical mappings and show the error between the analytically and computationally calculated mappings.

7 HOMEWORK # 2: FINITE DIFFERENCES MAPPING AND TWO DIMENSIONAL PROBLEMS 7 3. Question # 3: Stokes or Creeping Flow in a Cavity 2-Dimensional Rectangular Grid Finite Differences Note : This problem is mathematically the easiest, but is likely the most challenging of the three from a Matlab coding perspective partly because of the staggered grids used as well as the PDE being s system of three equations. In this problem you are tasked to solve a classical viscous flow fluid dynamics problem the Stokes flow for a classical test case, the cavity driven flow. In a cavity driven flow the top boundary has a unit x-velocity while the other boundaries are set to have a no slip condition ( v = 0). The domain is assumed to have unit dimensions in each direction. A staggered grid is employed in the MAC method we use to solve this problem. This means it will be convenient to setup ghost nodes outside of the domain to help with prescribing boundary conditions at the actual boundary. This is especially relevant for the pressure values. Figure 4. The domain in which the Stokes flow will be solved. The fluid domain is shown in blue shading. The boundary conditions are shown. The governing equations for incompressible fluid dynamics are the incompressible Navier- Stokes equations. In this case, the system of three equations comprise the x and y

8 8 DUE: WEDS MARCH 26TH 2018 momentum equations and the continuity or conservation of mass equation: ( u ρ t + u u x + v u ) = P ( 2 ) y x + µ u x u (5) y 2 + f x ( v ρ t + u v x + v v ) = P ( 2 ) y y + µ v x v (6) y 2 + f y u (7) x + v = 0 y We will assume that the flow is steady and the viscous terms dominate flow behavior. When we simplify the Navier Stokes equations we arrive at the steady state Stokes or creeping flow equations which are described as a system of three linear PDEs: ( 2 ) u (8) µ + P (9) (10) µ x u y 2 ( 2 v x v y 2 ) + P y u x + v y x = f x = f y = 0 To simplify matters, we assume that the fluid has no volume forcing terms (f x = 0 and f y = 0). We also assume for simplicity that the viscosity µ = 0. The result is a slightly simpler system of partial differential equations: ( 2 ) (11) u x u y 2 + P x = 0 ( 2 ) v x v (12) y 2 + P = 0 y u (13) x + v = 0 y For this problem we use a Marker and Cell (MAC) finite difference equation approach to discretize the equations. The MAC method discretizes the velocities at the midpoints of cell faces and the pressure at the center of the cells as shown in figure 5. Based on these grids, the equations can be discretized using finite difference expressions as shown in figure 6 (based on the grid shown). The grid spacing is assumes to be uniform in all directions, i.e., x = y = h. The x-momentum equation once discretized becomes: (14) 4u i,j u i,j 1 u i,j+1 u i 1,j u i+1,j x 2 + p i,j p i,j 1 x = 0

9 HOMEWORK # 2: FINITE DIFFERENCES MAPPING AND TWO DIMENSIONAL PROBLEMS 9 Figure 5. The three different meshes or grids used in the MAC solution method. (left) the u-velocity (velocity in the x-direction) is determined at vertical midpoint locations of the base grid (center) The v-velocity (velocity in the y-direction) is determined at the horizontal midpoint locations of the base grid (right) The pressure is determined at the cell center locations. The result is a grid that is offset in each direction. The y-momentum equation becomes: (15) 4v i,j v i,j 1 v i,j+1 v i 1,j v i+1,j x 2 + p i+1,j p i,j x = 0 The continuity or conservation of mass equation is: (16) u i,j u i 1,j. + v i,j v i,j 1 x x The boundary conditions are applied with the assistance of the ghost nodes on the outside of the domain. = 0 Please answer the following questions (hand-in a paper copy of answers, code will be uploaded via turnitin). (1) Show/write/draw how you will relate the i, j index to the global node number to each discrete equation (i.e., each node of the u, v and p grids). Note, you may wish to number the nodes/equations such that they directly correspond to the equation rows in the A-Matrix. (2) Show/write/draw how/where each of the three governing equations will be assembled to form an A-matrix.

10 10 DUE: WEDS MARCH 26TH 2018 (3) Indicate how you will implement the boundary conditions. Note, you will need to prescribe the pressure at one point to be a known value - the suggestion is to set pressure equal to zero on one of the boundary nodes. (4) Write the pseudo code for this problem implementation. (5) Write the Matlab code and compare the results of the Matlab code to those shown in figure 7. (6) Demonstrate the error convergence for your code. What is the accuracy of your code with respect to the grid refinement? (7) Bonus: Modify your code to accommodate curved/stretched grids. Calculate the viscous flow around a circle. You may find it most convenient to simulate a half circle, ie. the top of bottom half of a circle. Compare your result with the theoretical solution for the Stokes flow over a sphere.

11 HOMEWORK # 2: FINITE DIFFERENCES MAPPING AND TWO DIMENSIONAL PROBLEMS 11 Figure 6. The three equations apply to the grids shown. (left) The x- momentum equation (center) The y-momentum equation (right) The conservation of mass equation.

12 12 DUE: WEDS MARCH 26TH 2018 Figure 7. The solution to the cavity driven stokes flow equation. (top left) The magnitude of velocity in the cavity (top right) The velocity field in the cavity (bottom left) The x-velocity as measured on the vertical centerline (bottom right) The y-velocity as measured on the horizontal centerline.