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Partial Differential Equation Toolbox

Partial Differential Equation Toolbox
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Partial Differential Equation Toolbox - 1

Partial Differential Equation Toolbox Solve partial differential equations using finite element methods The Partial Differential Equation Toolbox™ product contains tools for the study and solution of partial differential equations (PDEs) in two-space dimensions (2-D) and time. A set of command-line functions and a graphical user interface let you preprocess, solve, and postprocess generic 2-D PDEs for a broad range of engineering and science applications. Key Features ▪ Complete GUI for pre- and post-processing 2-D PDEs ▪ Automatic and adaptive meshing ▪ Geometry creation using constructive solid geometry (CSG) paradigm ▪ Boundary condition specification: Dirichlet, generalized Neumann, and mixed ▪ Flexible coefficient and PDE problem specification using MATLAB syntax ▪ Fully automated mesh generation and refinement ▪ Nonlinear and adaptive solvers handle systems with multiple dependent variables ▪ Simultaneous visualization of multiple solution properties, FEM-mesh overlays, and animation Working with the Partial Differential Equation Toolbox The Partial Differential Equation Toolbox lets you work in six modes from the graphical user interface or the command line. Each mode corresponds to a step in the process of solving PDEs using the Finite Element Method. ▪ Draw mode lets you create Ω, the geometry, using the constructive solid geometry (CSG) model paradigm. The graphical interface provides a set of solid building blocks (square, rectangle, circle, ellipse, and polygon) that can be combined to define complex geometries. ▪ Boundary mode lets you specify conditions on different boundaries or remove subdomain borders. ▪ PDE mode lets you select the type of PDE problem and the coefficients c, a, f, and d. By specifying the coefficients for each subdomain independently, you can represent different material properties. ▪ Mesh mode lets you control the fully automated mesh generation and refinement process. ▪ Solve mode lets you invoke and control the nonlinear and adaptive solver for elliptic problems. For parabolic and hyperbolic PDE problems, you can specify the initial values and obtain solutions at specific times. For the eigenvalue solver, you can define the interval over which to search for eigenvalues. ▪ Plot mode lets you select from different plot types, including surface, mesh, and contour. You can simultaneously visualize multiple solution properties using color, height, and vector fields. The FEM mesh can be overlaid on all plots and shown in the displaced position. For parabolic and hyperbolic equations, you can animate the solution as it changes with time.

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Partial Differential Equation Toolbox - 2

Hie Edit Options Urs.w Boundary PUt Mesh i0W? Put Using the graphical user interface to define the complex geometry of a wrench, generate a mesh, and analyze it for a given load configuration. Accelerating the pace of engineering and science

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Partial Differential Equation Toolbox - 3

Defining and Solving PDEs With the Partial Differential Equation Toolbox, you can define and numerically solve different types of PDEs, including elliptic, parabolic, hyperbolic, eigenvalue, nonlinear elliptic, and systems of PDEs with multiple The basic scalar equation of the toolbox is the elliptic PDE where Vis the vector (^ffaidfSy)^ anj cjs a 2-by-2 matrix function on £2 , the bounded planar domain of interest, c, a, and /can be complex valued functions of xand y. Parabolic, Hyperbolic, and Eigenvalue PDEs The toolbox can also handle the parabolic PDE where <iis a complex valued...

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Partial Differential Equation Toolbox - 4

on where is the outward unit normal and g, q, h, and r can be complex valued functions of x and y defined on . For the nonlinear PDE, the coefficients may depend on u. For time-dependent problems, the coefficients may also depend on t. For PDE systems, Dirichlet, generalized Neumann, and mixed boundary conditions are supported. Visualization tools provide multiple ways to plot results. A contour plot with gradient arrows shows the temperature and heat flux. The temperature gradient is displayed using 3-D plotting tools. Toolbox Application Modes The Partial Differential Equation Toolbox...

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Partial Differential Equation Toolbox - 5

▪ Conductive Media DC ▪ Heat Transfer ▪ Diffusion The boundary conditions are altered to reflect the physical meaning of the different boundary condition coefficients. The plotting tools let you visualize the relevant physical variables for the selected application. Resources Product Details, Examples, and System Requirements www.mathworks.com/products/pde Online User Community www.mathworks.com/matlabcentral Trial Software www.mathworks.com/trialrequest Training Services www.mathworks.com/training Sales www.mathworks.com/contactsales Third-Party Products and Services...

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