Assignments are noted on the ES 551 schedule. Basic Matlab m-files are provided for introductory labs. More detailed labs require user-alteration of given Matlab m-files. Prepare your reports using the provided templates. Limit discussion to a few paragraphs with tabular data and plots included as appropriate. Clearly correlate conclusions with specified lab objectives. The Archived Lab is opened the following class period. Student reports can be accessed at Student Labs Late labs will not be accepted !
1.	State an n-dimensional PDE, with appropriate BCs, describing an engineering analysis problem of your interest. Briefly describe features of its solution prompting your interest, and anything you know about available solution processes (for the simplified or real problem). Prepare your report using the provided template, and mount it in your lab directory. Due at class 3.
2.	Complete the 1-D heat transfer linear basis convergence study, summarized in VuG HT1.10, see also Table HT1.2. Verify template construction forming the [DIFF] matrix, hence modifications for HBC and Dirichlet BCs. Realizing Texact=1000.00, verify the error relationship (HT1.16). Complete the report template. Due at class 8.
3.	Extend the GWSh template for the 1-D heat conduction convergence study to 1  k  3 FE bases, see VuG HT1.20, for uniform discretizations h, h/2, etc. Verify the asymptotic error estimates in the energy semi-norm and in Tmax using the slope equations (HT1.38) and (HT1.44). Due at class 9.
4.	Execute a convergence study for the conduction with distributed source problem, for a single material, then for two dissimilar conductivity materials with double sine source, VuG HT1.21, for 1  k  2 bases. Add code for computation of boundary fluxes from GWSh solution. Measure asymptotic convergence in ||Th||E and in Tmax using your data, hence compare to theory (HT1.38) and (HT1.45). Due at class 11.
5.	The Newton algorithm template for the non-linear GWSh for heat transfer with temperature-dependent conductivity is summarized on VuG HT1.24. For the conductivity linear temperature variation, execute a mesh refinement study measuring asymptotic convergence in Tmid and in ||Th||E norm for the k=1, 2 basis algorithms. Generate the Matlab template modifications for the conductivity quadratic temperature variation case, then repeat the mesh asymptotic convergence experiments. Finally, for this conduction variation and one mesh M, compare iteration convergence rates with and without the non-linear jacobian contribution. Due at class 13.
6.	Execute a convergence study, in max displacement and energy, for the Euler-Bernoulli beam problem using the cubic Hermite element formulation. Edit the Matlab template to create the resolved variable formulation, VuG CM1.9, using the k=1,2 bases, including the option for variable moment of inertia I(x). Select a suitable distribution for I(x), and a non-symmetric mix of distributed and point loadings, then confirm solution adherence to asymptotic convergence theory hence the mesh needed for 0.1% accuracy of max displacement. Due at fall break start..
7.	This fun problem corresponds to a dielectric heat source embedded in a 2-D heat sink. Matlab mesh generation is via Delauney triangulation of a macro element definition, yielding a versatile, non-uniform meshing capability. Review the Matlab template script, then generate GWSh solutions for a range of two material conductivities for the given Dirichlet BCs. Plot the results in planform and perspective. Due at class 17.
8.	A verification problem is radial heat conduction in a pipe flowing hot fluid. The analytical solution is radially logarithmic, hence all approximate solutions will have quantifiable error. Review the Matlab template script for the analysis description given in VuG HTn.18. Complete a GWSh natural coordinate k=1 basis solution mesh convergence study for regular azimuthal and regular radial discretization refinements for the given Robin and Dirichlet BCs. Compare surface temperature incremental accuracy improvement to the Chapter HT1 study problem solutions. Due at class 18.
9.	The L-shaped region with uniform heat source is a classic 2-D benchmark problem with solution character dominated by boundary geometric regularity. Verify the Matlab template script for the natural coordinate and tensor product k=1 basis GWSh implementations. Execute a uniform mesh refinement convergence study for either the TP (tensor product)  or NC (natural coordinate) basis implementation, expressing results graphically in the max and energy norms. Then fill in the omitted region, repeat the experiment, and establish conclusions about the influence of "data" smoothness. Due at class 20.
10.	For the unconfined aquifer, VuG.HTn.31, the conductivity k in the laplacian PDE is replaced by the dependent variable q, recall (PS.32)as the general statement. Thus, the aquifer PDE is genuinely non-linear which means the templates (HTn.66) or (HTn.67) are inappropriate, as the solution for GWSh must be written in Newton iterative form.  Recalling (CM1.71) - (CM1.72) for guidance, express the algetraic solution statement for GWSh in non-linear form for general dimension n. Then edit (HTn.66) and (HTn.67) to the associated template statements. Post your theoretical development and Newton templates as the report. Due at class 22.
11.	Execute a convergence study in the energy norm for the steady "Peclet problem," VuG CD1.10, for Pe=1 and Pe=10. Self-generate the required Matlab template (from Lab 3) for the k=1,2 basis algorithms. Increase Peclet number to Pe=100,1000, observe the solutions, then adapt h to a M=20 non-uniform meshing for improved solution accuracy using the "eye-ball" norm. Due at class 24.
12.	Execute the GWSh Matlab template script for the axisymetric unsteady heat conduction problem, VuG CD1.20. Using progression-ratio non-uniform mesh refinements, generate the Robin BC surface temperature accurate to within 0.1° for t-t0=0.001 hr. Start with t=0.0001 hr and decrease as determined necessary. Repeat a coarse mesh case using the =1.0 Euler time algorithm, hence quantify the impact of time truncation error on Tmax. Due at class 26.
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