Isogeometric Analysis On Implicit Domains

Numerical methods are an indispensable component for boosting the development in various scientific fields.The Finite Element Method(FEM)is considered the bench-mark numerical technique when dealing with problems related to partial differential equations that arise for example in engineering applications.However,the transla-tion from newly developed CAD designs to analysis is not automatic and comprise a bottleneck.An improvement that attracted attention in this direction surfaced when Hughes and coworkers proposed Isogeometric Analysis(IGA),which is a development of the standard FEM.IGA utilize CAD exact representation techniques including mainly NURBS basis functions for finite element analysis.Nevertheless,NURBS CAD models may require a high number of control points to satisfy the topological features for some sophisticated geometries due to their tensor product structure.In IGA,parameteriza-tion is a straightforward process for geometries that are represented using one NURBS patch.Still,parameterization is challenging when multiple patches are required to rep-resent complex geometries.Such obstacles motivate an attentive assessment of present techniques and calls for new developments in this field.This dissertation introduces Isogeometric Analysis on Implicit Domains,which is a new method that conducts analysis on geometries defined by implicit form splines.The new method innovatively blends the advantages of both IGA and the WEB method.The isogeometric concept is extended to incorporate implicit representations by adopting the same CAD basis functions that define the geometry in implicit form to conduct the anal-ysis.The implicit representation preserves the exact geometry and eliminates the need for a challenging parameterization step.Additionally,implicit geometries contain prop-erties,such as inside/outside point detection,which is crucial for modern applications in Additive Manufacturing,mainly for 3D printing.The method is carried out as a grid-based approach in order to overcome some of the complications associated with implicit representations such as the global control problem.Local refinable splines that usually applied in geometric modeling in CAD are further used as basis functions to provide solution spaces with higher continuity.Also,the weighted extended basis structures presented by the WEB method are updated and generalized to provide effective and flexible employment of different spline basis on grids.In order to overcome the parametrization problem,a solution structure based on weighted extended PHT-splines is constructed to solve second order partial equations.Unlike NURBS,PHT-splines allow local adaptive refinement naturally and can repre-sent the exact geometry in implicit form with an acceptable number of elements.More-over,the constructed basis functions with cubic polynomials and only C1 continuity are enough to produce a higher continuous field approximation while maintaining the computational cost for the matrices as low as possible.Several 2D numerical examples on implicit curved and free-form domains are presented.Furthermore,a numerical im-plementation for an application in quantum mechanics that include solving eigenvalue problems is included.The numerical results show that the method has good conver-gence.The numerical solution for higher order partial differential equations is challeng-ing in FEM due to the requirement of finite elements with higher regularity conditions.Another problem is the stability of the basis functions near the boundaries when ap-plying adaptive refinement for grid-based approaches.In this dissertation,a solution structure for weighted extended THB-splines is constructed to solve higher order partial differential equations using standard Galerkin weak form.In particular,a level-by-level classification mechanism applied to the spline basis.Consequently,leading to produce a hierarchical extension that connects basis functions with small support in the compu-tational domain to stable basis inside the domain.The basis functions are formulated to acquire local adaptive refinement and preserve stability to release the full approxima-tion power.Numerical implementation is performed to solve the Biharmonic equation on 2D implicit domains with holes and reentrant corners.Finally,the possibility of employing the new approach to additive manufacturing is examined,and preliminary numerical experiments in 3D are conducted to test the convergence results for solids constructed using implicit trivariate B-splines.The nu-merical results reflect the potential and capabilities of the new method.