MATLAB file only.Please use all code for part 5, 6, 7.Please follow the instructions carefully.
MXB222 Case Study Modelling an Electronic Component Due: Friday 26 October at 11:59pm 1 Background Your group works for a small start-up technology company in Brisbane. Part of your business is designing custom electronic components for embedded devices. In this case study, your group has been asked to determine the steady-state heat distribution in a newly designed component, to ensure it will perform to specifications. You will develop a mathematical model of the heat distribution in the component, solve it numerically using MATLAB, and prepare a report with your results to present to the company. 2 Component A schematic of the component is shown in Figure 1. The location of this component within the device means it is subject to different temperature conditions along its boundaries. Boundary A–B is in perfect thermal contact with another component whose temperature is known to be 70 C. Boundary C–D is in perfect thermal contact with another component whose temperature is known to be 40 C. Boundary A–E–D is thermally insulated. Boundary B–C is exposed to the air at ambient temperature. 1 0 0.01 0.02 0.03 0.04 0.05 0.06 0 0.01 0.02 0.03 0.04 0.05 0.06 x y A B C DE Figure 1: Schematic of electronic component 3 Model Steady-state heat conduction in the absence of sources or sinks is described by Laplace’s equation. Let T (x, y) represent the temperature of the component at point (x, y). The mathematical model for this component is as follows: ∂2T ∂x2 + ∂2T ∂y2 = 0 in the interior T = 70 on boundary A–B T = 40 on boundary C–D ∇T · n̂ = 0 on boundary A–E–D k∇T · n̂ = h(T∞ − T ) on boundary B–C The thermal conductivity k = 3 W m−1 C−1, and heat transfer coefficient h = 20 W m−2 C−1 are known. To begin, we suppose the ambient temperature is T∞ = 20. 4 Numerical discretisation You are required to discretise the model equations using the usual second order 5-point finite difference stencil in the interior, with first order finite difference approximations for boundary conditions. In this way, the resulting matrix will be symmetric positive definite. The discretisation may be carried out entirely by hand, with the resulting matrix and right hand side vector entered into MATLAB manually, or it may be completely automated using code, or something in-between. In any case, you must provide all your working. 2 5 Storage and solution You are required to write MATLAB code to perform the following tasks. Full storage Generate / load / import the coefficient matrix (as a standard two-dimensional array) and the right hand side vector. Factorise and solve the matrix system using your own implementation of the Cholesky algorithms (do not use MATLAB’s built-in chol function). Packed storage Generate / convert the coefficient matrix using packed storage. Factorise and solve the matrix system using your own implementation of the Cholesky algorithm for packed storage. Band storage Generate / convert the coefficient matrix with RCM ordering using band storage. You may use MATLAB’s symrcm function. Factorise and solve the matrix system using your own implementation of the Cholesky algorithms for band storage. Sparse storage Generate / convert the coefficient matrix with AMD ordering using MATLAB’s sparse storage. You may use MATLAB’s sparse and symamd functions. Factorise and solve the matrix system using your own implementation of the Cholesky algorithms for sparse storage (do not use MATLAB’s built-in chol function). CSR storage Generate / convert the coefficient matrix (with its original ordering) using CSR storage. Solve the matrix system using your own implementation of Jacobi, Gauss-Seidel, SOR and Conjugate Gradient algorithms for CSR storage. Visualisation Use visualisation methods to depict the temperature field graphically. 3 6 Efficiency comparison The company is interested in knowing which numerical method or methods provide the great- est efficiency in solving problems of this nature. You have been asked to investigate this, by performing the following tasks. Determine the bandwidth and level of fill-in for each different node ordering. Record and exhibit relevant efficiency measures: memory (for direct methods), itera- tions (for iterative methods), runtime (direct and iterative), floating point operations (direct and iterative). This must include a figure which plots the residual norm against iterations for the four iterative methods. Determine what tolerance is required for the iterative methods to produce a solution that is visually indistinguishable from that produced by the direct methods. Investigate the effect of ω on the rate of convergence of the SOR method. Include a plot of ω against the number of iterations required to converge to some chosen tolerance. Does the formula for the optimal value of ω given in the notes appear to hold for your matrix? 7 Effect of ambient temperature At the point (0.03, 0.03) on the component is a delicate piece of electronics that can only func- tion properly at temperatures between 50 C and 55 C. The company wants to market this component globally, so it must be sure it functions properly at all reasonable ambient tem- peratures. Determine what range of ambient temperatures this component can safely operate in, such that the temperature at the point (0.03, 0.03) remains between 50 and 55 C. On that basis, what are your recommendations regarding the global distribution of this component? 8 What to submit Report You must submit your findings in a report. Refer to http://www.citewrite.qut. edu.au/write/report.jsp for information on the structure of a report, with the exception that for this report it is not necessary to include a literature review. There is no set length for the report, but as a guide, 10 pages is too short and 50 pages is too long. The report must use appropriate formal language with correct spelling and grammar. It must be submitted in PDF format, which may be generated using LATEXor Microsoft Word, for example (but not MATLAB’s publishing feature, which is not suitable for a report of this nature). Code You must submit all your code, in the form of MATLAB functions and a single MATLAB script that calls your functions and exhibits your results. All MATLAB figures and statistics used in the report must be exhibited by this script. Spreadsheet A detailed spreadsheet of the components expected for the code and report, with their associated marks, has been provided. In this spreadsheet, you must record the group members responsible for undertaking each component. Group meeting summaries Summaries of your group meetings, which may include apolo- gies, agendas, action plans, outcomes, etc. 4 http://www.citewrite.qut.edu.au/write/report.jsp http://www.citewrite.qut.edu.au/write/report.jsp 9 Final mark The criteria for marking this piece of assessment are provided on the Assessment Task Descrip- tion document. This document should be read in conjunction with the provided spreadsheet, where detailed breakdown of marks for code and report are given. Generally a single overall mark will be awarded to the entire group. However, in cases where there is a genuine issue with the group work, for example, if one or more group members are ill or absent, consideration can be applied. It is essential that supporting documentation in the form of group meeting summaries are provided in order for consideration to be applied. Note that in the case of non-contributing group members, the remaining group members should not increase their workload to get the assessment completed. Instead, the work should be submitted in incomplete form, with the spreadsheet and group meeting summaries clearly indicating the allocation of workload and the issues that arose. 5 Background Component Model Numerical discretisation Storage and solution Efficiency comparison Effect of ambient temperature What to submit Final mark