First Derivatives Of The Approximation For Discontinuous Functions

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1.4.3 Discontinuities in derivatives Another important class of discontinuities are those in the first derivatives of the approximation. These discontinuities occur at interfaces between materials and different 13 1.4. APPROXIMATION FOR DISCONTINUOUS FUNCTIONS [? ] phases of materials. Discontinuities in derivatives of solutions occur wherever the coefficients of the governing partial differential equation are discontinuous. These discontinuities can easily be handled by standard finite element approximations by aligning the element edges with the discontinuity. However, if the discontinuity moves with time, remeshing is required. The approximation given below can model discontinuities in the derivatives on surfaces (or lines in 2D) which…show more content…
For example, for a vector function u(x), such as a displacement, a discontinuity on f (x) is introduced by u(x) =åi Ni(x)  ui+aiH( f (x))  : where ai is a column matrix of the same dimension as ui. The construction of discontinuities of a single component in a vector function is simplified by the use of the signed distance function. The unit normal to the line of discontinuity is given by en = Ñf kÑf k : 14 1.5. NUMERICAL EXAMPLE: ONE DIMENSIONAL BI-MATERIAL BAR [? ] Although a signed distance function should have a unit gradient, we normalize it here since this should be done in a computation. The tangent plane is then defined by any two unit vectors orthogonal to en. We illustrate the construction of the approximation in 2D. The discontinuity in the tangential component is obtained by letting the displacement field in the elements cut by the discontinuity be given by u(x) =åi Ni(x)  ui+aiet(x)H( f (x))  : where et =ezen is a vector in the tangent direction. Only a single parameter is needed at each node. 1.5 Numerical Example: One dimensional bi-material bar [40] In this section, the XFEM is illustrated with example involving weak discontinuities (material interfaces) to introduce the reader
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