Many ways to apply boundary conditions

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Boundary Conditions

There are many ways to apply boundary conditions in a element simulation. The way a boundary condition gets written depends totally on the way the weak problem has been designed; boundary conditions will be written quite differently in least-squares formula than in Galerkin formulation.

General considerations

Specifying where a boundary condition gets applied

àIn this, geometric subdomains are identifed using objects in the equations. The surface on which a BC is to be applied is specified by passing as an argument the representing to that surface.

àNonlinear Boundary Conditions

àSome problems will have nonlinear boundary conditions, for example in radiative heat transfer from a convex surface the heat flux is T4 Sundance requires you to handle nonlinear BCs in the same way you handle nonlinear PDEs: by iterative solution of a linearized problem. The linearization will depend on the iterative method used.

Nonlocal Boundary Conditions

àNonlocal boundary conditions arise naturally in problems such as radiative heat transfer from non-convex surfaces, and can also occur as far-field boundary conditions in acoustics, electromagnetics, or fluid mechanics.

à Nonlocal boundary conditions are handled with integral equations, and are not supported in the current version of Sundance

àMixed boundary value problems are characterised by a combination of Dirichlet and Neumann conditions along at least one boundary.

àHistorically, only a very small subset of these problems could be solved using analytic series methods (''analytic'' is taken here to mean a series whose terms are analytic in the complex plane).

àIn the past, series solutions were obtained by using an appropriate choice of axes, or a co-ordinate transformation to suitable axes where the boundaries are parallel to the abscissa and the boundary conditions are separated into pure Dirichlet or Neumann form. In this paper, I will consider the more general problem where the mixed boundary conditions cannot be resolved by a co-ordinate transformation.

àThat is, a Dirichlet condition applies on part of the boundary and a Neumann condition applies along the remaining section.

àI will present a general method for obtaining analytic series solutions for the classic problem where the boundary is parallel to the abscissa. In addition, I will extend this technique to the general mixed boundary value problem, defined on an arbitrary boundary, where the boundary is not parallel to the abscissa.

àI will demonstrate the efficacy of the method on a well known seepage problem

Mixed boundary condition

Green: Neumann boundary condition; purple: Dirichlet boundary condition.

àIn mathematics, a mixed boundary condition for a partial differential equation indicates that different boundary conditions are used on different parts of the boundary of the domain of the equation.

àFor example, if u is a solution to a partial differential equation on a set O with piecewisesmooth boundary ?O, and ?O is divided into two parts, G1 and G2, one can use Dirichlet boundary condition on G1 and a Neumann boundary condition on G2:

Robin boundary condition is another type of hybrid boundary condition; it is a linear combination of Dirichlet and Neumann boundary conditions.

Mixed Boundary Conditions

àThe allowed set of boundary conditions for Laplace's equation (or the Helmholtz equation) include Dirichlet or Neumann conditions, or a mixture in which we have Dirichlet on part of the boundary and Neumann on part.

àThese latter kinds of problems with "mixed" boundary conditions are more tricky. Let's look at an example.

àA thin conducting plane has a circular hole of radius a in it. We may choose coordinates so that the conducting plane is the x - y plane, with origin at the center of the hole.

àThe boundary conditions at z ? -8 are a uniform field E = E0 = E0ˆz. We want to solve for the fields everywhere. The boundary conditions on the plane are F = constant for ? > a

Ez is continuous for 0 < ? < a.


&agrave;In mathematics, the Robin (or third type) boundary condition is a type of boundary condition, named after Victor Gustavo Robin (1855-1897) who lectured in mathematical physics at the Sorbonne in Paris and worked in the area of thermodynamics.

&agrave; When imposed on an ordinary or a partial differential equation, it is a specification of a linear combination of the values of a function and the values of its derivative on the boundary of the domain.

&agrave; Robin should be pronounced as a French name , although some English-speaking mathematicians anglicize the word .

&agrave;Robin boundary conditions are a weighed combination of Dirichlet boundary conditions and Neumann boundary conditions.

&agrave;This contrasts to mixed boundary conditions, which are boundary conditions of different types specified on different subsets of the boundary.

-&agrave;Robin boundary conditions are also called impedance boundary conditions, from their application in electromagnetic problems.

&agrave;If O is the domain on which the given equation is to be solved and ?O denotes its boundary, the Robin boundary condition is:

for some non-zero constants a and b and a given function g defined on ?O. Here, u is the unknown solution defined on O and ?u/?n denotes the normal derivative at the boundary. More generally, a and b are allowed to be (given) functions, rather than constants.

notice the change of sign in front of the term involving a derivative: that is because the normal to [0, 1] at 0 points in the negative direction, while at 1 it points in the positive direction.

&agrave;Robin boundary conditions are commonly used in solving Sturm-Liouville problems which appear in many contexts in science and engineering.

&agrave;In addition, the Robin boundary condition is a general form of the insulating boundary condition for convection-diffusion equations. Here, the convective and diffusive fluxes at the boundary sum to zero:

Dirichlet boundary condition

&agrave;In mathematics, the Dirichlet (or first-type) boundary condition is a type of boundary condition, named after Johann Peter Gustav Lejeune Dirichlet When imposed on an ordinary or a partial differential equation, it specifies the values a solution needs to take on the boundary of the domain.

The question of finding solutions to such equations is known as the Dirichlet problem.

&agrave;Dirichlet boundary conditions are perhaps the easiest to understand but there are many other conditions possible.

&agrave;For example, there is the Cauchy boundary condition or the mixed boundary condition which is a combination of the Dirichlet and Neumann conditions.