# 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.

### Specifying where a boundary condition gets applied

&agrave;In 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.

### &agrave;Nonlinear Boundary Conditions

&agrave;Some 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

&agrave;Nonlocal 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.

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

&agrave;Mixed boundary value problems are characterised by a combination of Dirichlet and Neumann conditions along at least one boundary.

&agrave;Historically, 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).

&agrave;In 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.

&agrave;That is, a Dirichlet condition applies on part of the boundary and a Neumann condition applies along the remaining section.

&agrave;I 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.

&agrave;I will demonstrate the efficacy of the method on a well known seepage problem

### Mixed boundary condition

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

&agrave;In 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.

&agrave;For 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

&agrave;The 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.

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

&agrave;A 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.

&agrave;The 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.

### &egrave;ROBIN BOUNDARY CONDITION

&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.

### BOOKS:-

NUMERICAL ANALYSIS BY M.K JAIN.

B.S GREWAL

SITES:-