1 Exercise: Heat equation in 2-D with FE
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1 1 Exercise: Heat equation in 2-D with FE Reading Hughes (2000, sec Dabrowski et al. (2008, sec. 1-3, 4.1.1, 4.1.3, This FE exercise and most of the following ones are based on the MILAMIN package by Dabrowski et al. (2008 which provides a set of efficient, 2-D Matlab-based FE routines including a thermal and a Stokes fluid solver. Given that the code uses Matlab, MILAMIN is remarkably efficient and certainly a good choice for simple 2-D research problems that lend themselves to FE modeling. You may want to consider working on expanding the MILAMIN capabilities, e.g. by adding advection to the thermal solver and combining it with the Stokes solver for a convection code. Over the next sections, we will discuss all of the issues described in Dabrowski et al. (2008. This paper will be a good additional reference, and the original MILAMIN Matlab codes can be downloaded from (the latter will not be of help with the exercises. 1.1 Implementation of 2-D heat equation We spent the last three sections discussing the fundamentals of finite element analysis building up to the solution of the 2-D, stationary heat equation, which is given by ( κ T + ( κ T = H, (1 x x dz where κ is conductivity (not diffusivity, we use κ to distinguish from the stiffness matrix K, and H are heat sources. Both κ and H may vary in space, and, unlike for FD, the solution domain can now be irregular. The FE approach casts the boundary value problem (boundary conditions are assumed given in the weak (variational form, discretized on elements on which shape functions, N, approximate the solution of the PDE as T. The solution is given by nodal temperatures T = {T A } for all NNOD nodes of the mesh, which can be combined to T(x, z = NNOD A=1 N A (x, zt A. (2 Following, e.g., Hughes (2000, we use the Galerkin approach for which the resulting stiffness matrix components, on an element level, is K e ab = Ω e κe ( Na x USC GEOL557: Modeling Earth Systems 1 x + N a dω. (3
2 Here, a and b are node numbers local to element e, and integration Ω e is over the element area. If we express the spatial coordinates x = {x, z} in a node-local coordinate system ξ = {ξ, η} and use Gaussian quadrature with NINT points and weights W i for integration, we need to evaluate terms of the kind K e ab = K e ab = 1 1 dξ 1 1 dηκ ( Na + N a NINT ( Na N W i κ b i + N a i J 1 j (4 J 1 j (5 where J 1 is the inverse and j = det(j = J the determinant of the Jacobian matrix J = ( x x, (6 respectively. The load vector F has to be assembled on an element basis as Fa e = N ahdω K e ˆT Ω e ab b, (7 where the terms on the right hand side are due to heat sources, H, and a correction due to prescribed temperatures on the boundaries ˆT (zero flux BCs need no specific treatment, see Hughes, 2000, p. 69 and Dabrowski et al. (2008. The global K and F are assembled by looping through all elements and adding up thek e and F e contributions, while eliminating those rows that belong to nodes where essential boundary conditions ( ˆT are supplied. The solution is then obtained from solving Meshing KT = F. (8 Download generate_mesh.m. Start by reading through this Matlab code, it is a modification of the MILAMIN wrapper for triangle. Triangle is a 2-D triangular mesh generator by Shewchuk (2002. That work is freely available as C source code and a flexible, production quality Delaunay mesh generator. A Delaunay mesh is such that all nodes are connected by elements in a way that any circle which is drawn through the three nodes of an element has no other nodes within its circumference. Aside: The dual graph (sort of the graphic opposite of a Delaunay mesh are the Voronoi cells around each node. Those can be constructed based on the triangulation by connecting lines that are orthogonal to each of the triangles sides and centered half-way between nodes. Those two properties are important for computational geometry, inverse theory, and interpolation problems. USC GEOL557: Modeling Earth Systems 2
3 A Delaunay triangulation is the best possible mesh for a given number of nodes in the sense that the triangles are closest to equilateral. For FE analysis, we always strive for nicely shaped elements (i.e. not distorted from their ideal, local coordinate system form so that the J 1 does not go haywire, and j = det(j remains positive. Typically, meshers like triangle will allow you to refine the mesh (i.e. add more nodes for a given boundary structure and overall domain by enforcing minimum area and/or angle constraints. Those refinements may also be iteratively applied based on an initial solution of the PDE, e.g. to refine in local regions of large variations (adaptive mesh refinement, AMR. Exercise Download a test driver for the triangle wrapper, mesher_test.m. You will have to fill in the blanks after reading through generate_mesh.m, and make sure the triangle binary (program is installed on your machine in the directory you are executing your Matlab commands in. Note: For this exercise and those below, please first inspect graphs on the screen while playing with the code, and then only print out a few geometries. 1. Create a triangular grid using three node triangles for the domain 0 x 1, 0 z 1 using minimum area constraint 0.1 and minimum angle 20. Create a print out plot of this mesh highlighting nodes that are on the outer boundary. 2. Change the area constraint to 0.01, remesh, and replot. 3. Use second order triangles and an area constraint of and minimum angle of Using the same quality constraints, create and print out a mesh plot of an elliptical inclusion of radius 0.2, ellipticity 0.8, and 50 nodes on its perimeter. Color the elements of the inclusion differently from those of the exterior. Denote nodes on the boundary of the inclusion. 5. Create and plot a mesh with a circular hole and a circular inclusion of radius Thermal solver Downloadthermal2d_std.m; this is a simplified version of the MILAMIN thermal solver (thermal2d.m which should be easier to read than the version of Dabrowski et al. (2008; it also allows for heat production. Read through this Matlab code and identify the matrix assembly and solution method we discussed in sec Please make sure you take this step seriously. Also download and read through shp_deriv_triangle.m and ip_triangle.m which implement linear (three node and quadratic (six node, triangular shape functions and derivatives, and weights for Gauss quadratures, respectively. USC GEOL557: Modeling Earth Systems 3
4 Exercise 1. Download a rudimentary driver for the mesher and thermal solver, thermal2d_test.m. You will need to fill in the blanks. 2. Generate a regular mesh with area constraint and solve the heat equation with linear shape functions, without heat sources, given no flux on the sides, unity temperature at the bottom, and zero temperature at the top. Plot your results. Use constant conductivity. 3. Place an elliptical inclusion with radius 0.4, ellipticity 0.8, and ten times higher conductivity than the ambient material in the medium, and plot the resulting temperatures. Experiment with variable resolutions and second order triangles. Comment on the how the solution changes (visually only is OK. 4. Set the heat production of the inclusion to 10 and 100, and plot the solution. Compare with boundary conditions where zero temperatures are prescribed on all boundary conditions. 5. Compute the temperature as well as the geothermal gradient at a specific location. This exercise is so that you gain experience using shape functions and derivatives of shape functions and requires you to identify the N and N equivalents. USC GEOL557: Modeling Earth Systems 4
5 BIBLIOGRAPHY Bibliography Dabrowski, M., M. Krotkiewski, and D. W. Schmid (2008, MILAMIN: MATLAB-based finite element method solver for large problems, Geochem., Geophys., Geosys., 9(Q04030, doi: /2007gc Hughes, T. J. R. (2000, The finite element method, Dover Publications. Shewchuk, J. R. (2002, Delaunay refinement algorithms for triangular mesh generation, Comput. Geom.: Theor. Appl., 22, USC GEOL557: Modeling Earth Systems 5
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