In the finite element simulation operation, the solution to the PDE equation is often encountered. The commonly used simulation software such as COMSOL, ANSYS, and ABAQUS are tools for the PDE equation. In order to solve the PDE equation to have a definite solution, it is necessary to introduce a certain condition - the definite solution condition, sometimes called the singularity condition. In order to determine the solution of the equation, it is necessary to provide sufficient initial and boundary conditions.
(1) Initial value condition
If the equation requires the unknown quantity y(x) and its derivative y'(x) to take a given value at the same point x=x0 of the independent variable, that is, y(x0)=y0, y'(x0)=y0', then This condition is called the initial condition, and the problem composed of the equation and the initial condition is called the initial value problem;
(2) Boundary conditions
In many practical problems, the solutions of differential equations are often required to satisfy certain conditions at the endpoints of a given interval a ≤ x ≤ b, such as y(a) = A, y(b) = B, then give The condition of the value at the end point (boundary point) is called the boundary condition
The boundary conditions are described in detail below. There are various forms of boundary conditions in boundary value problems, which can be roughly written as such at the endpoints, Ay+By'=C:
- If B=0, A≠0, it is called the first-type boundary condition or Dirichlet condition, which gives the value of the unknown function on the boundary ;
- If B≠0, A=0, it is called the second type of boundary condition or Neumann condition, which gives the directional derivative of the unknown function outside the boundary ;
- If A≠0, B≠0, it is called the third kind of boundary condition or Robin condition, which gives the linear combination of the function value of the unknown function on the boundary and the derivative of the outer normal .
The simulation software I use now is the multiphysics simulation COMSOL, and in COMSOL there are only two kinds of boundary conditions:
- Dirichlet boundary (the first kind of boundary condition), at the endpoint, the value of the variable to be evaluated is specified.
- Neumann boundaries (type 2 boundary conditions), which are often given in the form of Flux, where the directional derivative of the normal outside the boundary of the variable to be found is specified.
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