8,943 research outputs found
Finite element methods for surface PDEs
In this article we consider finite element methods for approximating the solution of partial differential equations on surfaces. We focus on surface finite elements on triangulated surfaces, implicit surface methods using level set descriptions of the surface, unfitted finite element methods and diffuse interface methods. In order to formulate the methods we present the necessary geometric analysis and, in the context of evolving surfaces, the necessary transport formulae. A wide variety of equations and applications are covered. Some ideas of the numerical analysis are presented along with illustrative numerical examples
Multilevel Solvers for Unstructured Surface Meshes
Parameterization of unstructured surface meshes is of fundamental importance in many applications of digital geometry processing. Such parameterization approaches give rise to large and exceedingly ill-conditioned systems which are difficult or impossible to solve without the use of sophisticated multilevel preconditioning strategies. Since the underlying meshes are very fine to begin with, such multilevel preconditioners require mesh coarsening to build an appropriate hierarchy. In this paper we consider several strategies for the construction of hierarchies using ideas from mesh simplification algorithms used in the computer graphics literature. We introduce two novel hierarchy construction schemes and demonstrate their superior performance when used in conjunction with a multigrid preconditioner
Numerical modelling of heat transfer and experimental validation in Powder-Bed Fusion with the Virtual Domain Approximation
Among metal additive manufacturing technologies, powder-bed fusion features
very thin layers and rapid solidification rates, leading to long build jobs and
a highly localized process. Many efforts are being devoted to accelerate
simulation times for practical industrial applications. The new approach
suggested here, the virtual domain approximation, is a physics-based rationale
for spatial reduction of the domain in the thermal finite-element analysis at
the part scale. Computational experiments address, among others, validation
against a large physical experiment of 17.5 of deposited
volume in 647 layers. For fast and automatic parameter estimation at such level
of complexity, a high-performance computing framework is employed. It couples
FEMPAR-AM, a specialized parallel finite-element software, with Dakota, for the
parametric exploration. Compared to previous state-of-the-art, this formulation
provides higher accuracy at the same computational cost. This sets the path to
a fully virtualized model, considering an upwards-moving domain covering the
last printed layers
Optimal Control Of Surface Shape
Controlling the shapes of surfaces provides a novel way to direct
self-assembly of colloidal particles on those surfaces and may be useful for
material design. This motivates the investigation of an optimal control problem
for surface shape in this paper. Specifically, we consider an objective
(tracking) functional for surface shape with the prescribed mean curvature
equation in graph form as a state constraint. The control variable is the
prescribed curvature. We prove existence of an optimal control, and using
improved regularity estimates, we show sufficient differentiability to make
sense of the first order optimality conditions. This allows us to rigorously
compute the gradient of the objective functional for both the continuous and
discrete (finite element) formulations of the problem. Moreover, we provide
error estimates for the state variable and adjoint state. Numerical results are
shown to illustrate the minimizers and optimal controls on different domains
Development of an unsteady aerodynamic analysis for finite-deflection subsonic cascades
An unsteady potential flow analysis, which accounts for the effects of blade geometry and steady turning, was developed to predict aerodynamic forces and moments associated with free vibration or flutter phenomena in the fan, compressor, or turbine stages of modern jet engines. Based on the assumption of small amplitude blade motions, the unsteady flow is governed by linear equations with variable coefficients which depend on the underlying steady low. These equations were approximated using difference expressions determined from an implicit least squares development and applicable on arbitrary grids. The resulting linear system of algebraic equations is block tridiagonal, which permits an efficient, direct (i.e., noniterative) solution. The solution procedure was extended to treat blades with rounded or blunt edges at incidence relative to the inlet flow
Calibrations and isoperimetric profiles
We equip many non compact non simply connected surfaces with smooth
Riemannian metrics whose isoperimetric profile is smooth, a highly non generic
property. The computation of the profile is based on a calibration argument, a
rearrangement argument, the Bol-Fiala curvature dependent inequality, together
with new results on the profile of surfaces of revolution and some hardware
know-how.Comment: To appear soon in American Journal of Mathematics, a journal
published by The Johns Hopkins University Pres
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