Elements of MATLAB and Simulink - Lecture 7 - PERCRO

Elements ofMatlab and SimulinkLecture 7Emanuele Ruffaldi12th May 2009Copyright 2009,Emanuele Ruffaldi. This work is licensed under the Creative Commons Attribution-ShareAlikeLicense.

PARTIAL DIFFERENTIALEQUATIONS

Introducing PDEOrdinary Differential Equation• Derivatives respect a single independent variable• Solution is a function with arbitrary constantsPartial Differential Equation• Differential relationship of multiple variables• Solution is an arbitrary function• Solution determined by fixing boundary conditionsNotation

Elements of a PDEEquationDomainBoundaryConditions

Heat EquationHeat (or diffusion) equation: models the diffusion of temperaturefrom an initially concentrated distributionu_t = u_xxExample of solution is:u(x,t) = 1/sqrt(t) exp(-x^2/4*t)Verify using MATLAB Symbolic Toolbox>> Example lecture7_diffuse

Boundary ProblemThe Dirichlet problem is the general problem of finding a function uwhich solves a PDE for which the values are known on theboundary of a given region• Given a PDE over R N• Given a function f that has values over a boundary region of Ω• Find a function u solving PDE that is differentiable twice in theinterior and once on the boundary and assumes the values of f onthe boundaryThe Neumann problem involves a boundary condition relative to aderivative of the target functionExpressed as u_n

Heat Equation 2DHeat Equation 2D: diffusion of a quantity along the space and timeu_t = u_xx + u_yyThe general formulation isu_t = div grad uExampleMetal block with a rectangular crack. The left side is heated at 100degrees, the right side heat is flowing to air at constant rate. Theothers sides are insulatedu = 100 (Dirichlet condition)u_n = -10 on the right side (Neumann condition)u_n = 0 on the other boundaries (Neumann condition)

Linear Advection EquationLinear advection equation: models the constant movement of aninitial distribution of u with a speed of –c along x axis. Shape ispreservedu_t = c u_xFor any function q a solution is q(w) where w = x+ctu(x,t) = q(w) = q(x+ct)>> Example lecture7_advection

Wave EquationLinear Waveu_tt = c^2 u_xxA typical solution is u(x,t)=sin(x+ct)Spherical Waveu_tt = c^2 (u_rr + 2/r u_r)If we put in evidence (ur) the solution is the same as aboveu(t,r) = 1/r [F(r-ct)+G(r+ct)]>> Example lecture7_wave

Laplace EquationLaplace Equationu_xx + u_yy = 0Poisson Equationu_xx + u_yy = g(x,y,z)Helmholt’s Equationu_xx + u_yy + f(x,y) u = g(x,y,z)Require boundary conditions for resolution

Basics of FEMHow we can solve PDE numerically when no analytical solution isavailable?The Finite Element Method is the most common solution1. Decompose the Domain in subspaces2. Approximate the Function with piecewise linear function definedinside each subspace3. Find the numerical solution for every subspace given theconditionsFDM is the Finite Difference Method is possible but it works betterwith rectangular domains

FEM 1DSpace discretizationInterpolation Function Linear Combination (Basis)

FEM 2DAirSiliconGeneralized approach by decomposingthe domain in unit elements

Families of PDE and use• Elliptic and Parabolic• Steady and unsteady heat transferin solids• Flows in porous media anddiffusion problems• Electrostatics of dielectric andconductive media• Potential flow• Hyperbolic• Transient and harmonic wavepropagation in acoustics andelectromagnetics• Transverse motions of membranes• Eigenvalue• Determining natural vibrationstates in membranes andstructural mechanics problems

MATLAB Partial Differential Equation ToolboxSolves some families of PDE: elliptics, parabolic, hyperbolic andeigenvaluePDE Toolbox works in the easiest way with 2D BoundaryUser Interface (pdetool) and Command lineSteps1. Define the Domain2. Define the Boundary function3. Solve4. Plot results

Expressing the GeometryThe boundary is expressed by Constructive Solid GeometrySet composition of fundamental entitiesThe CSG allows to compose basic entities1 32 4-Graphical editing (pdetool)-initmesh and decsg-Understand CSG expression

Geometry as Mesh• The FEM resolution uses a triangular mesh for the spatialdiscretization of the problem• The Constructive Geometry produces a closed surface that is latertransformed in Mesh• Refine mesh for improving details on borders or surfacefeatures (number of triangles increases)• Jiggle mesh (quality improves)Draw ModeMesh Mode

Expressing the Boundary Conditions• Two types of Boundary Conditions• Dirichlet• h u=r• Neumann• n c grad(u) + qu = g• e.g. n c grad(u) = 0 means no flux• Coefficients in bold can be specified with PDE tool• Export Decomposed geometry and Boundary into Workspacevariables

Resolution of PDE• Solver function depend on the type of problem• u1=parabolic(u0,tlist,b,p,e,t,c,a,f,d)• u0 is the initial condition (sized as size(p,2))• tlist is the time in which let system evolve• b boundary• p,e,t geometry• [c,a,f,d] parameters of• Automatic Refinement of problem• [u,p,e,t]=adaptmesh(g,b,c,a,f,options)• The parabolic equation can be solved keeping time fixed andstarting at given condition• Time Discretization (stages)• Method of lines• Allows PDEs in Simulink

Heat Equation 2DExampleMetal block with a rectangular crack. The left side is heated at 100degrees, the right side heat is flowing to air at constant rate. Theothers sides are insulatedu = 100 (left side)u_n = -10 on the right sideu_n = 0 on the other boundariesu_t = u_xx + u_yyCreate DomainSet Boundary conditionsSet PDESolveTest>> Example lecture7_heat2d_pde.m with pdetool

PDE for Simulink• Simulink Solves ODE equations• Use Method of Line for making the PDE an ODE problem• We are interested in using the dynamic resolution of a PDE as partof our simulation• Update the PDE parameters and inputs along time• Integration performed by Simulink• If using Discrete simulation we can let MATLAB do the mathStatesOne for every nodePDEProblemSimulinkBlockInputTime dependingOutputsStatitsics (max/min)Value at node

PDE in Simulink PracticalWe use S-Functions Level-2Block InitializationThe output is the state of the PDE that depends on the number ofnodes of the GeometryDerivativesUpdate the Derivatives from the Method of LinesOutput ComputationReturn statistics about the states, and eventually return sampledvalues from some nodes

Matrix Form of the FEMBasic Matrix FormulationDirichletNeumannIntegrated Equation

Thermal Regulation Problem• If the diffusion is fast there is noneed for space information• What if we want to use the exactposition of the Thermostat?PDE is ParabolicSimulink just needs the Elliptic

Thermal Regulation PDE• Geometry• Boundary• Walls: n grad(T)=0• Glass: n grad(T)=q (u(1)-T)• Where u(1) is exterior temperature, T is the internal temperature and q is the thermal conductivity• Equation• T' = div( K grad(T) ) + Th u(2)• u(2) is the heater on or off• Th is the function of heat flow from the heater based on its position

Thermal Regulation SimulinkThe solution for this integration is an S-Function written as M-fileParameters• Coordinates of Heater xh,yh and Radius• Node of the Thermostat• Initial Temperature T0• Heater Temperature ThInput• State of the Heater• External TemperatureOutput• State of the Heater• Maximum Temperature in Room• Temperature at Thermostat

References• Books• An Introduction to Partial Differential Equations withMATLAB,Coleman M., CRC Press, 2005• Introduction to Partial Differential Equations with MATLAB,Cooper J.M., Birkhause, 1998• Beginners Guide to Simulation for Physical Engineering:Practical Usage of MATLAB and PDEase, Yutaka Abe, 1999• Courses• Introduction to Partial Differential Equations, Doug Moore,2003• Introduction to Partial Differential Equations, Showalter,Oregon State University• MAE502 Partial Differential Equations in Engineering,ASU,2009