Finite Element Analysis ME 452w

	Course Calendar details computer lab assignments. Theory validation labs employ Matlab using .m files accessible here. Design study labs employ the COMSOL Multi-Physics Problem Solving Environment via .mph files accessible here. Prepare your reports using the template zip file download on the PSE page. Keep discussion concisely focused on generated tabular data and graphics. Clearly correlate Conclusions with Objectives. The Lab Archive is opened one day after assignment. Student reports are accessed at Student Labs Late computer lab reports will not be accepted !

1.	Identify an engineering analysis problem statement of your specific interest. Briefly describe the features that prompt your interest and anything you know about available solution processes. Prepare your report using the provided lab template and mount it at your public_html directory. Due at class 4.

2	The complete step by step weak statement process, linear finite element basis implemented, was detailed for an elementary 1-D steady DE problem in Chapter 4. Number crunching was not involved as the the development retained data names throughout. The conclusion was that analytically exact answers were generated at the nodes of any mesh, but that error was clearly present in any FE approximate solution. The algorithm template is summarized in Table 5.2. Download its Matlab .m file, below, and input the numbers for the BCs and other data pertinent to defining a real problem. Conduct a regular mesh refinement study using factors of two, i.e., M = 2, 4, 8, 16, 32, hence generate the data necessary to verify the theoretical asymptotic convergence prediction in the energy norm, (5.43). Use the report template to fill in the Discussion section with pertinent tabular data and a convergence graph, then state your Conclusions with reference to the stated Objectives. Downloads: lab2.m Due at class 7.

3.	(Lab 6.5) Small horsepower air-cooled gasoline engines achieve thermal balance via fins molded integral to the cylinder. Via the lecture, a simplified DE principle generates a 1-D axi-symmetric ODE with a radiation BC at the cylinder wall and convection BCs on the fin surfaces. Download this problem statement template Matlab .m file. Input the BCs and other data specific for this analysis given in the report template. Conduct a mesh refinement study to determine the M necessary to predict the cylinder wall temperature to an accuracy of one-half degree. Prepare your report Discussion, including appropriate graphs and tabular data, and complete the Conclusions with specific reference to Objectives. Downloads: lab3.m Due at class 9.

4.	(Lab 11.3) Heat transfer in practical large-scale situations typically involves fluid flowing over banks of horizontal tubes. Buoyancy effects can exert a profound impact on this process. Following the text discussion, also instructions in the lab report template, execute the COMSOL .mph file case for the base Re and Gr definitions. Then determine the inflow velocity data change yielding Gr/Re^2 for heat transfer mode alteration. Note the issue with outflow BC for the larger Re specification, hence fix the problem. Compute the average Nu associated with each heat transfer mode. Prepare your report Discussion, including appropriate graphs and tabular data, and complete the Conclusions with specific reference to Objectives. Downloads: COMSOL 3.5a: natural_coarseMesh_35a.mph \|\| natural_constants.txt Due at class 11.

5.	(Lab 6.2) The structural mechanics introductory lecture presented two forms of dP leading to 1-D ODE pairs for prediction of the deflection of horizontal beams due to loading and various BCs. Download the GWS template .m files for the Euler-Bernoulli and Timoshenko beam theory algorithm statements. Conduct mesh refinement studies for both beam formulations using M refinement by factors of two, hence determine the meshing necessary to predict the maximum deflection to engineering accuracy. Confirm asymptotic rates of convergence in the energy norm. Alter the loading to a mixture of distributed and point loads and repeat the convergence study. Prepare your report Discussion, including appropriate graphs and tabular data, and complete the Conclusions with specific reference to Objectives. Downloads: lab5eb.m \|\| lab5t.m Due at class 13.

6.	(Lab 11.1) In structural mechanics, the plane stress/strain assumption for linear, elastic, homogeneous media leads to forms for dP or DE amenable to GWS^h hence FE basis discrete implementations. This design study problem aks that you adjust the shape of an initially-circular hole in a plate under tension, assuming plane stress, to moderate the maximum local stress concentration. Execute the COMSOL .mph file base case, then a solution-adapted regular mesh refinement to estimate mesh adequacy. Recognizing the wall BC divergence constraint issue, alter the BCs as directed and repeat the mesh refinement. Then adjust the shape of the circular hole, keeping the area constant, such that the extremum von Mises stress is minimized for the entire plate. Prepare your report Discussion, including appropriate graphs and tabular data, and complete the Conclusions with specific reference to Objectives. Downloads: COMSOL 3.5a: circular_normalMesh_35a.mph \|\| elipse_normalMesh_35a.mph Due at class 16.

7.	(Lab. 6.3 edit) Access the Euler-Bernoulli beam theory normal mode GWS^h formulation Matlab .m file. Input the BCs and other data specific for the cantilever beam lecture example. Generate the resulting first four natural frequencies and associated normal mode shapes for M = 4 to 64. Estimate the uniform mesh M required to obtain engineering accuracy for each natural frequency. Since the displacement solutions are smooth, estimate the natural frequency convergence rate order of accuracy. Prepare your report Discussion, including appropriate graphs and tabular data, and complete the Conclusions with specific reference to Objectives. Downloads: lab7.m Due at class 19.

8.	(Lab 11.4) Multi-dimensional structural mechanical vibrations analyses are usually cast in normal mode form, hence DP is transformed to an eigenvalue problem statement. For vibration of an L-shaped membrane, execute the COMSOL base case solving for the first ten eigenvalues and eigenmodes. Execute a mesh refinement study to assess natural frequency accuracy. Then place a fillet in the membrane concave corner, regenerate the mesh and the corresponding solution and compare eigenvalues/eigenmodes to the base case. Draw conclusions on the action of the fillet in modifying the lowest few normal mode shapes and frequencies. Prepare your report Discussion, including appropriate graphs and tabular data, and complete the Conclusions with specific reference to Objectives. Downloads: COMSOL 3.5a: basefillet_35a.mph \|\| sharpcorner_35a.mph Due at class 21.

9.	(Lab 6.4) Aerodynamics weak interaction theory combines a far-field potential flow formulation of DM with laminar/turbulent boundary layer forms for DM and DP. This exercise seeks solutions to quasi-1D potential flow in a venturi device. Access the Matlab .m file and execute the base case specification, then conduct a mesh refinement convergence study for phi, p and u. Prepare your report Discussion, including appropriate graphs and tabular data, and complete the Conclusions with specific reference to Objectives. Downloads: lab9.m Due at class 22.

10.	(Lab 11.2 edit) The classic Navier-Stokes validation is flow in a duct with sudden enlargement in cross-section, the step-wall diffuser. In 2-D, the NS DM and DP transformation to streamfunction-vorticity enables a mathematically well-posed PDE + BCs system. The design goal is to establish the primary re-circulation bubble intersection point with the lower wall, normalized by step height, as a function of 100 <= Re <= 600. This coordinate is defined by the wall vorticity DOF distribution sign change. Conduct a solution-adapted mesh refinement student for the Re range, paying close attention to validity of the outflow BC for larger Re. Prepare your report Discussion, including appropriate graphs and tabular data, and complete the Conclusions with specific reference to Objectives. Downloads: COMSOL 3.5a: baseMesh_35a.mph Due at class 24.

11.	(Lab 9.1 edit) Non-dimensional groups precisely categorize term dominance in scalar heat/mass transport statements. The need for solution adaptive meshing is verified for the n=1 steady DE "Peclet problem". Download the Matlab .m file, hence determine the uniform meshing required to generate oscillation-free GWS^h solutions for Pe = 1, 10, and 100. Then find the least non-uniform M = 20 mesh supporting k = 1,2 basis monotone solutions for Pe = 100 and 1000. (Lab 9.2 edit) Edit the Lab 9.1 .m file to support a k = 1 basis study of the unsteady traveling energy packet problem, (9.33), for 1/Pe = 0. Interpolate the IC onto a uniform M = 20 mesh, hence compute the number of time steps required for the analytical solution to travel exactly two IC wavelengths at C = 0.5. Repeat this exercise for C = 1.0. Then alter the theta = 0.5 definition to theta = 0 and any theta > 0.5 and compare impact on solution accuracy. Edit the .m file to implement the k = 2 basis template and repeat Prepare your report Discussion, including appropriate graphs and tabular data, and complete the Conclusions with specific reference to Objectives. Downloads: lab11a.m \|\| lab11b.m Due at class 25.

12.	(Lab 10.4 edit) The multi-dimensional extensions on the Lab 11 DE and DM studies are the n = 2 Peclet and rotating cone statements. The steady Peclet problem facilitates direct GWS^h and optimal gamma mGWS^h theory comparison regarding dispersion error control with enhanced asymptotic convergence rate. The rotating cone clearly assesses GWS^h and optimal gamma mGWS^h algorithm performance for 1/Pa vanishingly small as fully documented in the lecture. Download the Matlab .m file for the n=2 Peclet problem GWS^h algorithm. Edit the template to implement the optimal gamma k = 1 basis mGWS^h algorithm. Perform a regular mesh refinement study for uniform M = 10x10 to 160x160 meshes for Pe = 10. Plot asymptotic convergence rate in the E norm and comment on the data. Then repeat the regular M refinement process for Pe = 1000 and graph the mGWS^h theory asymptotic convergence rate in the E norm. Prepare your report Discussion, including appropriate graphs and tabular data, and complete the Conclusions with specific reference to Objectives. Downloads: lab12.m Due at class 27.

13.	(Lab 11.5 edit) A multi-D practical multi-physics simulation of solid-fluid mass transfer/transport is the subject. Access the COMSOL .mph file and execute the base case specification. Then adjust the effective diffusion coefficient D to alter the solid diffusion level, hence also the Schmidt number. Run a series of simulations varying the data definitions over the ranges 1E-06 < D < 9 E-06, 5 < Re < 72, hence 4 < Sc < 31 and observe solution distinctions. Perform a solution adapted regular mesh refinement for the largest Re and Sc specification being particularly attentive to control of dispersion error oscillations just downstream of the pellet major diameter. Prepare your report Discussion, including appropriate graphs and tabular data, and complete the Conclusions with specific reference to Objectives. Downloads: COMSOL 3.5a: expSetup_35a.mph Due at class 28.

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