MATLAB finite differences

FIGURE 3.13 Solution of the DPM isothermal drying model of one-dimensional plate by pdepe solver of MATLAB . Finite difference discretization by uniform mesh both for space and time, 5. is dimensionless time, xlL is dimensionless distance. 

A few additional remarks with respect to the calculation of the Jacobian matrix J are in order here. For reaction mechanisms that have explicit solutions to the set of differential equations, it is always possible to define the derivatives dC/dk explicitly. In such cases, the Jacobian J can be calculated in explicit equations, and time-consuming finite-difference approximations are not required. The equations are rather complex, although implementation in MATLAB is straightforward. More information on this topic can be found in the literature [22], The calculation of numerical derivatives is always possible, and for mechanisms that require numerical integration it is the only option. 

For the dynamic lung impedance model to be useable in Finite Difference Method or Finite Element Method impedance signal simulations, the dynamic tissue sample model is discretized into volume data. At first 3D data with 35 x 35 x 35 voxel resolution is prepared from each of the 40 time frames. This allows for easy import into MATLAB or COMSOL based calculation. The volume data includes percentage of blood vessels (blood) for each of the 35 X 35 X 35 X 40 voxels. It can readily be transformed into electric/dielectric properties for each voxel with tissue data available on the internet. But data can also be exported with arbitrary resolution depending on calculation-simulation requirements. The simulations are run separately for each of the 40 time-frames to get full frequency characteristic of impedance measurement across the tissue sample. Finally we can get 40 frequency characteristics—one for each time-frame and to see a dynamic electrical impedance signal on a certain frequency, we just need to plot the impedance value at the chosen frequency from the 40 time-frames. 

A MatLab code was created to solve equations 2 to 6 simultaneously. The constants in table 1 were used to generate a solution and the numerical methods apphed are Euler method and finite difference method. The finite difference is used to estimate the second order space differential and the forward Euler... 

The impedance simulation of the dynamics was carried out with Finite Difference Method (FDM) implemented in MATLAB. Alternatively, the discretization of equations can be performed by use of Finite Element Method (FEM), wavelets techniques, etc. [15,16]. Electrodes were positioned on 2 sides of the 10 mm square tissue model, current was inserted and voltages were calculated for each case of 60 time-steps. Electric impedance maps with resolution 35 x 35 x 1 were derived from the dynamic vascular network (Figure 4). 

The set of differential equations written above were solved numerical by the finite difference method of Runge-Kutta (Ode23tb of Matlab). Basically, a polymer sample is discretized into N knots equally spaced with a distance Ax ... 

Abstract Chapter 5 provides an examination of the numerical solutions of the dyeing models that can be applied to different conditions. Numerical simulation of the system involves the use of Matlab software to solve systems of highly non-linear simultaneous coupled partial differential equations. The finite difference and finite element methods are introduced The partition of the fibrous assembly geometry into small units of a simple shape, or mesh, is examined. Polygonal shapes used to define the element are briefly described. The defined geometries, boundary conditions, and mesh of the system enable solutions to the equations of flow or mass transfer models. 

In MATLAB, the function diff y) returns forward finite differences of y. Values of nth-order forward finite difference may be obtained from diffiy, n). 

Program Description The MATLAB function fder.m evaluates the derivative of a function. The first part of the program is initialization, where inputs to the function are examined and default values for differentiation increment (h) and method of finite difference are applied, if required. Introducing these two inputs to the function is optional. The program then switches to a different part of the program, according to the choice of method of finite difference, and then it switches to the proper section according to the order of differentiation. 

Develop the finite difference approximation of Pick s second law of diffusion in polar coordinates. Write a MATLAB program that can be used to solve the following problem [10] ... 

Convert this problem to dimensionless form to reduce the munber of independent parameters. Then, use the finite difference method to convert this boimdary value problem into a set of nonlinear algebraic equations and solve with MATLAB. Plot the dimensionless concentration as a function of the remaining adjustable dimensionless parameteifs). To speed up your calculations, have your function routine return the Jacobian matrix. 

Above, we have focused on solving BVPs by implementing the finite difference method directly. MATLAB also has a dedicated 1-D BVP solver, pdepe, for systems of equations of parabohc and elhptic type. Its use is rather straightforward, and for further details type doc pdepe. 

For static and (structural) dynamic analysis, for determination of eigenfre-quencies and eigenmodes, several different commercial tools exist such as NASTRAN, ABAQUS or ANSYS. Some of them are also able to handle actuators and piezoelectric materials, and also to carry out some types of model reduction techniques. Nevertheless, specific techniques might have to be established by the user via accessing the modal data base. These data are then also used to set up a modal or otherwise condensed state-space representation possibly including specific actuator and sensor models. A description of the transformation of finite-element models from ANSYS to dynamic models in state space form in MATLAB can be found in [20]. 

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