Discretizing PDEs for MapleSim

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Phillip Whitehead
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1 Discretizing PDEs for MapleSim Introduction MapleSim TM is a system-level modeling and simulation tool based on Modelica. MapleSim is integrated with Maple TM, a tool for symbolic and numeric analysis. Both tools can exchange information and data in either direction. Multidomain Modeling and Simulation Platform PDE Discretization & Analysis Platform MapleSim solves ODEs and DAEs. Hence PDEs must be discretized before simulation. The link with Maple means that many methods can be used to discretize PDEs and automatically generate Modelica-based components. The workflow is described in the following diagram: Step 1: Launch Maple from within MapleSim (custom component template) Step 2: Discretize PDE with Maple s functionality Step 3: Generate MapleSim component Step 4: Use component in MapleSim model

2 Maple can be used to implement different discretization schemes for PDEs, including: Analytical method Integral approximation method Pade approximation method Finite element method Finite difference method Method of lines Ritz method Galerkin s method (see. T.-S. Dao, et al., J. Power Sources (2011), doi: /j.jpowsour ). These techniques have been used to discretize PDEs for simulation in MapleSim, including: The PDEs governing a Lithium-ion battery using Galerkin s method; The coupled PDEs describing the temperature dynamics of a double-pipe heat exchanger; and The heat equation (and other basic PDEs) using a finite difference method. Section 2 describes how the heat equation is discretized with the finite difference method in Maple for simulation in MapleSim. Similar principles can be applied to more complex PDEs. Discretizing the Heat Equation 2.1 Launch the Custom Component Template Within MapleSim, 1. Go to View > Create Attachment 2. Select Custom Component Template 3. Click Create Attachment

3 2.2 Implementation of the Discretization Scheme in Maple In this section, items preceded with > are commands typed into Maple. This is the heat equation as entered in Maple. The input into the block is i_source(t). The single spatial derivative will now be automatically replaced with a central difference approximation. This can be done in Maple by (i) mechanised manipulation or (ii) programmatically. The following command programmatically replaces spatial derivatives with a central difference (see Appendix 1 for details). Only a single term needs is replaced; if present, other spatial derivatives would also be replaced. The index i represents a symbolic index. The entire set of ODEs is now generated by sweeping the discretized PDE from 1 to N, where N is the number of spatial discretizations. Hence the entire set of ODEs is:

4 Set u(0,x) and u(l+deltax,0): 2.3 Create MapleSim Component The inputs and outputs are now assigned to block ports with the Custom Component template. This is simply point-and-click. Size of each spatial step (L will be a block parameter): Hence the DEs to be implemented are: Set the initial temperature of all the state variables: Collect the equations, parameters and initial conditions for component configuration and automatic Modelica generation:

5 Now the MapleSim block can be created by clicking on Generate MapleSim Component. This automatically generates the Modelica source. The Modelica can be viewed and modified if desired. The block is now automatically available in MapleSim (in the Project->Definitions>Components menu). The block can be dragged into the workspace and connected to other components. The PDE or the degree of discretization can be changed by editing the custom component template associated with the component, making the appropriate alterations, and regenerating the component.

6 Key Contacts Dr Samir Khan Application Engineer Dr Than-Son Dao Battery modeling, PDE discretization, mathematical modeling Dr Juergun Gerhard Advanced discretization methods, symbolic algebra, mathematical modeling, PDE analysis Appendix This Maple procedure replaces spatial derivatives with a central difference approximation: disc:=proc(eqnum,disvarsset) local univariate,disvar,inputd: inputd := convert(eqnum, D); univariate:=inputd: for disvar in disvarsset do univariate := eval(univariate, [ disvar = ((x, t) -> disvar[i](t)), D[2](disVar) = ((x, t) -> D[1](disVar[i])(t)), D[2,2](disVar) = ((x, t) -> D[1,1](disVar[i])(t)), D[1](disVar) = ((x, t) -> (disvar[i+1](t)-disvar[i-1](t))/(2*deltax)), D[1, 1](disVar) = ((x, t) -> (disvar[i+1](t) - 2 * disvar[i](t) + disvar[i-1](t))/deltax^2),null]): end do: return convert(univariate,diff)[]: end proc: info@maplesoft.com Toll-free: (US & Canada) Direct: Maplesoft, Maple, and MapleSim are trademarks of Waterloo Maple Inc. Maplesoft, MapleSim, and Maple are trademarks of Waterloo Maple Inc. Modelica is a registered trademark of the Modelica Association.

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