The general theory of solutions to Laplace's equation is known as potential theory. In the study of heat conduction, the Laplace equation is the steady-state heat equation. In three dimensions, the problem is to find twice-differentiable real-valued functions , of real variables x, y, and z in 3D such as eq I.1.This is often written as eq. I.2 or eq. I.3. Where eq. I.3 is divergence of the function and eq. I.4 is known as its gradient. Also commonly depicted as Δ , the Laplace operator. The Laplace's and Poisson's equations are the simplest examples of elliptic partial differential equations. The partial differential operator, or (which may be defined in any number of dimensions), is called the Laplace operator or just the Laplacian. ---------------------------===================---------------------------------- Also known as the essential boundary condition. The Dirichlet boundary condition given for a an ordinary differential equation: where alpha and beta are given, always constants, and the function is defined in the [0,1] domain. ---------------------------===================---------------------------------- Also known as the natural boundary condition. The Neumann boundary condition is defined below:
Sparsness : However, examining ill. B.2, reveals that the products and nonzero only where the supports of for basic functions and , e.g. nonzero only on element 2, wheres zero everywhere. Hence, the integrals ,nonzero but . Then, it follows that if two nodes i and j not belong to the same element , then K[ij] =0. Symmetry of K : This property is easy to defend as we can see that interchanging the I and j integral expression for K[ij] not change the value calculated Yet in most physical problems based on conservation laws this symmetry will arise often naturally in the weak formulation. In contrast to the sparseness of K, the symmetry is independent of our choice of basis functions and it is entirely dependent on the type of variational problem we are trying to solve.
A memristor is a passive two-terminal electronic component for which the resistance (dV/dI) depends in some way on the amount of charge that has flowed through the circuit. When current flows in one direction through the device, the resistance increases; and when current flows in the opposite direction, the resistance decreases, although it must remain positive. When the current is stopped, the component retains the last resistance that it had, and when the flow of charge starts again, the resistance of the circuit will be what it was when it was last active
Petar Petrov MSc thesis defense
Towards predicting the effects of TMS electro-magnetic stimulation on the human brain Petar Petrov www.ppetrov.net ID#3196607
Research Goals <ul><li>Study the effects of TMS on the human brain using computer model and virtual simulation </li><ul><li>Choose a numerical solution suited for the physics of anisotropic ! electro-conductivity
Create anatomically correct computer model of a brain, incorporating the four most essential tissue types using MRI
Create, test and validate a working FEM solution </li></ul><li>Clinical motivation </li><ul><li>Improve future TMS lab session (aided neuronavigation) </li></ul></ul>
Theoretical Background <ul><li>Generalization of the Ohm's law for volume conductors
We might need to take different approach towards WM, than adaptive meshing </li></ul><li>Anisotropic effects </li><ul><li>Clear difference in patterns distribution </li><ul><li>Low thresholding used however (~5% of MAX E-field) </li></ul></ul><li>Resource bottle-neck (3xAMD64 6GB RAM) </li><ul><li>SCIRun is still interactive ~ 1M tet-elements
Meshing via BioMesh3D for 4tissue Brain ~ 3days! </li></ul><li>The error introduced through interpolation is relatively low! </li></ul>
Future Directions <ul><li>Empirical validation of our SCIRun FEM models </li><ul><li>TMS + EEG (low spatial distribution information)
TMS + fMEI (still experimental @ UMC Utrecht) </li></ul><li>Does Anisotropic WM modeling affects TMS clinical lab application ? </li><ul><li>We have shown a clear difference, but is it relevant?