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chap4_V3

# chap4_V3 - Chapter 4 Scalar Basis Functions and Application...

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Chapter 4 Scalar Basis Functions and Application in 2D In the last few chapters, we use several different basis functions with one spatial variable to solve 1-D problems and 2-D surface integral equations. The highest order we have used is linear. In many problems, higher order basis functions are advantageous. Such basis functions will be developed in this chapter. To simplify the discussion, we will restrict ourselves to scalar basis functions. Such basis functions are appropriate only for scalar fields or vector fields that only has one nonzero component, for example, the E z component in the 2-D TM z case, or the H z component in the 2-D TE z case. Such scalar basis functions are not appropriate for vector fields because they will enforce the continuity conditions for all field components. In reality, usually vector field do not have continuity in all components; enforcing such continuity introduces fictitious sources between adjacent elements. Vector basis functions for vector fields will be discussed later. 4.1 1-D Basis Functions 4.1.1 1-D Linear Basis Functions So far we have learned the pulse and triangular basis functions for 1-D functions, which are zeroth-order and first-order basis functions, respectively. We have represented the triangular basis functions associated with the nodal points as t n ( x ; x n - 1 , x n , x n +1 ), which are piecewise linear functions. Alternatively, we can also represent these functions in terms of their simplex coordinates within each element. Mathematically, “sim- plex” means a basic geometric element in a Euclidean space, for example, a line segment in one dimension, a triangle in two dimensions, and a tetrahedron in three dimensions. Simplex coordinates are also known as barycentric coordinates. Note that in 1D, each element is associated with two basis functions. The simplex coordinates ( s 0 , s 1 ) can be related to the Cartesian coordinates as x = s 0 x 0 + s 1 x 1 (4.1) where x 0 and x 1 are the locations of the end points of the element. Obviously, we have two equations for s 0 and s 1 : x 1 = x 0 x 1 1 1 s 0 s 1 (4.2) This equation gives s i = D i /D , where D = x 1 - x 0 and D i is the distance from the i + 1 node, using the cyclic notation ( i + 2 = i ). Equivalently, s 0 ( x ) = x 1 - x x 1 - x 0 , s 1 ( x ) = x - x 0 x 1 - x 0 (4.3) 65

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CHAPTER 4. SCALAR BASIS FUNCTIONS AND APPLICATION IN 2D Obviously, these two functions can be written more compactly as s i ( x ) = x - x i +1 x i - x i +1 , i + 2 = i (4.4) for i = 0, 1. In numerical implementation, it is more convenient to define the basis function on a reference element (or standard element) for - 1 ξ 1. The mapping between the physical space and the reference space is x = x 1 - x 0 2 ξ + x 0 + x 1 2 (4.5) Then the simplex coordinates can be written as s i ( ξ ) = ξ - ξ i +1 ξ i - ξ i +1 , i + 2 = i (4.6) where ξ 0 = - 1 and ξ 1 = 1. The integration of a function f ( x ) can be written as Z x 1 x 0 f ( x ) dx = Z 1 - 1 f ( x ( ξ )) det( J x ) (4.7) where J x = dx/dξ = ( x 1 - x 0 ) / 2 is the Jacobian matrix (in this case a scalar) of the mapping.
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