Type of Boundary Conditions

Hello Everyone, I wrote this article because it may be useful for Electromagnetic FEA simulation engineer who wants to know about boundary conditions in-depth.

Please correct me if i am wrong somewhere and give your comments.

Type of boundary condition and what is the reason for Neumann boundary conditions are natural in magneto static and electrostatic problem.

Consider the following one dimensional electrostatic Partial differential equation (PDE),

Domain is from 0<x<L.

Boundary condition: Drichlet V(0) = C, Neumann B.C εdv/dx at x=L is E.

In FEM, we use integral statement to develop algebraic relation among the coefficient’s of the approximations,

Reason for using the weighted integral statement:

·        When we substitute the approximation function (2) in to the governing equation (1), we do not get required number of linearly independent algebraic equation among the unknown coefficient Vi.

·        One way to ensure that there are exactly same number of equations as there are unknowns is to require weighted integrals of the error in the equation to be zero.

Step 1: Weighted Integral statement of PDE (1)  :

Equation (3) is called weighted residual statement equivalent to the original PDE. W is weight function.

Step 2: Weak form of Equation 3:

Advantage of weak form of (3) is differentiation is distributed among the approximate solution Vn and weight function w. Hence resulting integral from will require the weaker continuity condition on Ψi.

When we integrate the equation (3) by parts,

Equation (4) is called weak from of integral statement. Weak refers to the reduced continuity of V which is required to be twice differentiable in (3) but only once differentiable in (4).

Types of Boundary:

Examine all the boundary terms in equation 4. Here boundary terms are .

Secondary Variable: Coefficients of the weight function and its derivatives are termed as secondary variable. Specification of secondary variable on the boundary conditions are natural boundary condition.

Here coefficient of the weight function is εW(dv/dx) . So,boundary condition  at X=L is called natural boundary condition.

Primary Variable: Dependent variable of the problem expressed in the same form as weight function appearing in the boundary term is called primary variable. Here Form of weight function in boundary term is class 1 i.e w. Hence Primary variable in this case V. Specification of primary variable on the boundary conditions are essential boundary condition. Hence V at X=0 is called essential boundary condition.

 Step 3:

In weak formulations, weight function has the meaning of virtual change of the primary variable. If primary variable is specified at a point, virtual change there must be zero. Hence, W(0)=0 because V(0)=C.

Reference: Introduction to the Finite Element method by J N Reddy.