Abstract
In the paper [1], the geometrical mapping techniques based on Non-Uniform Rational B-Spline (NURBS) were introduced to solve an elliptic boundary value problem containing a singularity. In the mapping techniques, the inverse function of the NURBS geometrical mapping generates singular functions as well as smooth functions by an unconventional choice of control points. It means that the push-forward of the NURBS geometrical mapping that generates singular functions, becomes a piecewise smooth function. However, the mapping method proposed is not able to catch singularities emerging at multiple locations in a domain. Thus, we design the geometrical mapping that generates singular functions for each singular zone in the physical domain. In the design of the geometrical mapping, we should consider the design of control points on the interface between/among patches so that global basis functions are in C0 space. Also, we modify the B-spline functions whose supports include the interface between/among them. We put the idea in practice by solving elliptic boundary value problems containing multiple singularities.
Highlights
It has been introduced to solve multiple crack problems by using various numerical methods
In the design of the geometrical mapping, we should consider the design of control points on the interface between/among patches so that global basis functions are in C0 space
The potential of the mapping method proposed with multiple patches regarding to handling the multiple fatigue-cracks propagation in various types of plate will be shown by solving the elliptic boundary value problems with multiple singularities or cracks
Summary
It has been introduced to solve multiple crack problems by using various numerical methods. We solve the elliptic boundary value problems with multiple singularities based on the mapping method [1]. In the paper [11], we approximate the solution on the small circular zone centered at the crack tip or point singularity by enriching the finite approximation space generated by the singular mapping introduced in the mapping method. The potential of the mapping method proposed with multiple patches regarding to handling the multiple fatigue-cracks propagation in various types of plate will be shown by solving the elliptic boundary value problems with multiple singularities or cracks.
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