Search results for "DOMAIN"

Two-level Schwarz method for unilateral variational inequalities

1999

The numerical solution of variational inequalities of obstacle type associated with second-order elliptic operators is considered. Iterative methods based on the domain decomposition approach are proposed for discrete obstacle problems arising from the continuous, piecewise linear finite element approximation of the differential problem. A new variant of the Schwarz methodology, called the two-level Schwarz method, is developed offering the possibility of making use of fast linear solvers (e.g., linear multigrid and fictitious domain methods) for the genuinely nonlinear obstacle problems. Namely, by using particular monotonicity results, the computational domain can be partitioned into (mes…

A New Distributed Optimization Approach for Solving CFD Design Problems Using Nash Game Coalition and Evolutionary Algorithms

2013

For decades, domain decomposition methods (DDM) have provided a way of solving large-scale problems by distributing the calculation over a number of processing units. In the case of shape optimization, this has been done for each new design introduced by the optimization algorithm. This sequential process introduces a bottleneck.

A Visualizable Test Problem Generator for Many-Objective Optimization

2022

Visualizing the search behavior of a series of points or populations in their native domain is critical in understanding biases and attractors in an optimization process. Distancebased many-objective optimization test problems have been developed to facilitate visualization of search behavior in a two-dimensional design space with arbitrarily many objective functions. Previous works have proposed a few commonly seen problem characteristics into this problem framework, such as the definition of disconnected Pareto sets and dominance resistant regions of the design space. The authors’ previous work has advanced this research further by providing a problem generator to automatically create use…

A Domain Decomposition/Nash Equilibrium Methodology for the Solution of Direct and Inverse Problems in Fluid Dynamics with Evolutionary Algorithms

2008

Counting common perpendicular arcs in negative curvature

2013

Let $D^-$ and $D^+$ be properly immersed closed locally convex subsets of a Riemannian manifold with pinched negative sectional curvature. Using mixing properties of the geodesic flow, we give an asymptotic formula as $t\to+\infty$ for the number of common perpendiculars of length at most $t$ from $D^-$ to $D^+$, counted with multiplicities, and we prove the equidistribution in the outer and inner unit normal bundles of $D^-$ and $D^+$ of the tangent vectors at the endpoints of the common perpendiculars. When the manifold is compact with exponential decay of correlations or arithmetic with finite volume, we give an error term for the asymptotic. As an application, we give an asymptotic form…

Approximation by mappings with singular Hessian minors

2018

Let $\Omega\subset\mathbb R^n$ be a Lipschitz domain. Given $1\leq p<k\leq n$ and any $u\in W^{2,p}(\Omega)$ belonging to the little H\"older class $c^{1,\alpha}$, we construct a sequence $u_j$ in the same space with $\operatorname{rank}D^2u_j<k$ almost everywhere such that $u_j\to u$ in $C^{1,\alpha}$ and weakly in $W^{2,p}$. This result is in strong contrast with known regularity behavior of functions in $W^{2,p}$, $p\geq k$, satisfying the same rank inequality.

Pointwise inequalities for Sobolev functions on generalized cuspidal domains

2022

We establish point wise inequalities for Sobolev functions on a wider class of outward cuspidal domains. It is a generalization of an earlier result by the author and his collaborators

METRIC DIFFERENTIABILITY OF LIPSCHITZ MAPS

2013

AbstractAn extension of Rademacher’s theorem is proved for Lipschitz mappings between Banach spaces without the Radon–Nikodým property.

Fractional Hardy-Sobolev type inequalities for half spaces and John domains

2018

As our main result we prove a variant of the fractional Hardy-Sobolev-Maz'ya inequality for half spaces. This result contains a complete answer to a recent open question by Musina and Nazarov. In the proof we apply a new version of the fractional Hardy-Sobolev inequality that we establish also for more general unbounded John domains than half spaces.

Controlled diffeomorphic extension of homeomorphisms

2018

Let $\Omega$ be an internal chord-arc Jordan domain and $\varphi:\mathbb S\rightarrow\partial\Omega$ be a homeomorphism. We show that $\varphi$ has finite dyadic energy if and only if $\varphi$ has a diffeomorphic extension $h: \mathbb D\rightarrow \Omega$ which has finite energy.