A NUMERICAL METHOD FOR PROGRESSIVE LENS DESIGN

2004 ◽  
Vol 14 (04) ◽  
pp. 619-640 ◽  
Author(s):  
JING WANG ◽  
FADIL SANTOSA
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Numerical Method ◽  
Elliptic Partial Differential Equation ◽  
Computational Method ◽  
Necessary Condition ◽  
Perturbation Approach ◽  
Lens Design ◽  
Partial Differential ◽  
Nonlinear Elliptic

The problem of progressive lens design can be posed as a variational problem. The necessary condition is a fourth-order nonlinear elliptic partial differential equation. The partial differential equation can be linearized using a perturbation approach. A numerical method using a special type of splines, chosen for their smoothness properties, is devised to solve the resulting PDE. The computational method is shown to be both convergent and efficient.

The onset of multi-valued solutions of a prescribed mean curvature equation with singular non-linearity

2013 ◽  
Vol 24 (5) ◽  
pp. 631-656 ◽  
Author(s):  
N. D. BRUBAKER ◽  
A. E. LINDSAY
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Blow Up ◽  
Elliptic Partial Differential Equation ◽  
Classical Solutions ◽  
Necessary Condition ◽  
Partial Differential ◽  
End Point ◽  
Dead End ◽  
The Dead

The existence and multiplicity of solutions to a quasilinear, elliptic partial differential equation with singular non-linearity is analysed. The partial differential equation is a recently derived variant of a canonical model used in the modelling of micro-electromechanical systems. It is observed that the bifurcation curve of solutions terminates at single dead-end point, beyond which no classical solutions exist. A necessary condition for the existence of solutions is developed, revealing that this dead-end point corresponds to a blow-up in the solution's gradient at a point internal to the domain. By employing a novel asymptotic analysis in terms of two small parameters, an accurate characterization of this dead-end point is obtained. An arc length parameterization of the solution curve can be employed to continue solutions beyond the dead-end point; however, all extra solutions are found to be multi-valued. This analysis therefore suggests that the dead-end is a bifurcation point associated with the onset of multi-valued solutions for the system.

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