Elements of Matlab and Simulink Lecture 7

PARTIAL DIFFERENTIAL EQUATIONS

Introducing PDE Ordinary Differential Equation Derivatives respect a single independent variable Solution is a function with arbitrary constants Partial Differential Equation Differential relationship of multiple variables Solution is an arbitrary function Solution determined by fixing boundary conditions Notation

Elements of a PDE Equation Domain Boundary Conditions

Heat Equation Heat (or diffusion) equation: models the diffusion of temperature from an initially concentrated distribution u_t = u_xx Example of solution is: u(x,t) = 1/sqrt(t) exp(-x^2/4*t) Verify using MATLAB Symbolic Toolbox >> Example lecture7_diffuse

Boundary Problem The Dirichlet problem is the general problem of finding a function u which solves a PDE for which the values are known on the boundary 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 the interior and once on the boundary and assumes the values of f on the boundary The Neumann problem involves a boundary condition relative to a derivative of the target function Expressed as u_n

Heat Equation 2D Heat Equation 2D: diffusion of a quantity along the space and time u_t = u_xx + u_yy The general formulation is u_t = div grad u Example Metal block with a rectangular crack. The left side is heated at 100 degrees, the right side heat is flowing to air at constant rate. The others sides are insulated u = 100 (Dirichlet condition) u_n = -10 on the right side (Neumann condition) u_n = 0 on the other boundaries (Neumann condition)

Linear Advection Equation Linear advection equation: models the constant movement of an initial distribution of u with a speed of c along x axis. Shape is preserved u_t = c u_x For any function q a solution is q(w) where w = x+ct u(x,t) = q(w) = q(x+ct) >> Example lecture7_advection

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

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

Basics of FEM How we can solve PDE numerically when no analytical solution is available? The Finite Element Method is the most common solution 1. Decompose the Domain in subspaces 2. Approximate the Function with piecewise linear function defined inside each subspace 3. Find the numerical solution for every subspace given the conditions FDM is the Finite Difference Method is possible but it works better with rectangular domains

FEM 1D Space discretization Interpolation Function Linear Combination (Basis)

FEM 2D Air Silicon Generalized approach by decomposing the domain in unit elements

Families of PDE and use Elliptic and Parabolic Steady and unsteady heat transfer in solids Flows in porous media and diffusion problems Electrostatics of dielectric and conductive media Potential flow Hyperbolic Transient and harmonic wave propagation in acoustics and electromagnetics Transverse motions of membranes Eigenvalue Determining natural vibration states in membranes and structural mechanics problems

MATLAB Partial Differential Equation Toolbox Solves some families of PDE: elliptics, parabolic, hyperbolic and eigenvalue PDE Toolbox works in the easiest way with 2D Boundary User Interface (pdetool) and Command line Steps 1. Define the Domain 2. Define the Boundary function 3. Solve 4. Plot results

Expressing the Geometry The boundary is expressed by Constructive Solid Geometry Set composition of fundamental entities The CSG allows to compose basic entities 1 3 2 4 -Graphical editing (pdetool) -initmesh and decsg -Understand CSG expression

Geometry as Mesh The FEM resolution uses a triangular mesh for the spatial discretization of the problem The Constructive Geometry produces a closed surface that is later transformed in Mesh Refine mesh for improving details on borders or surface features (number of triangles increases) Jiggle mesh (quality improves) Draw Mode Mesh 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 Workspace variables

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 and starting at given condition Time Discretization (stages) Method of lines Allows PDEs in Simulink

Heat Equation 2D Example Metal block with a rectangular crack. The left side is heated at 100 degrees, the right side heat is flowing to air at constant rate. The others sides are insulated u = 100 (left side) u_n = -10 on the right side u_n = 0 on the other boundaries u_t = u_xx + u_yy Create Domain Set Boundary conditions Set PDE Solve Test >> 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 part of our simulation Update the PDE parameters and inputs along time Integration performed by Simulink If using Discrete simulation we can let MATLAB do the math States One for every node PDE Problem Simulink Block Input Time depending Outputs Statitsics (max/min) Value at node

PDE in Simulink Practical We use S-Functions Level-2 Block Initialization The output is the state of the PDE that depends on the number of nodes of the Geometry Derivatives Update the Derivatives from the Method of Lines Output Computation Return statistics about the states, and eventually return sampled values from some nodes

Matrix Form of the FEM Basic Matrix Formulation Dirichlet Neumann Integrated Equation

Thermal Regulation Problem If the diffusion is fast there is no need for space information What if we want to use the exact position of the Thermostat? PDE is Parabolic Simulink 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 Simulink The solution for this integration is an S-Function written as M-file Parameters Coordinates of Heater xh,yh and Radius Node of the Thermostat Initial Temperature T0 Heater Temperature Th Input State of the Heater External Temperature Output State of the Heater Maximum Temperature in Room Temperature at Thermostat

References Books An Introduction to Partial Differential Equations with MATLAB,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