scholarly journals hp-Version discontinuous Galerkin methods on polygonal and polyhedral meshes

2014 ◽  
Vol 24 (10) ◽  
pp. 2009-2041 ◽  
Author(s):  
Andrea Cangiani ◽  
Emmanuil H. Georgoulis ◽  
Paul Houston
Keyword(s):  
Discontinuous Galerkin ◽  
A Priori ◽  
Physical Space ◽  
Two Dimensions ◽  
Three Dimensions ◽  
Interior Penalty ◽  
Polyhedral Meshes ◽  
Canonical Frame ◽  
Polygonal Elements ◽  
Polynomial Bases

An hp-version interior penalty discontinuous Galerkin method (DGFEM) for the numerical solution of second-order elliptic partial differential equations on general computational meshes consisting of polygonal/polyhedral elements is presented and analyzed. Utilizing a bounding box concept, the method employs elemental polynomial bases of total degree p (𝒫p-basis) defined on the physical space, without the need to map from a given reference or canonical frame. This, together with a new specific choice of the interior penalty parameter which allows for face-degeneration, ensures that optimal a priori bounds may be established, for general meshes including polygonal elements with degenerating edges in two dimensions and polyhedral elements with degenerating faces and/or edges in three dimensions. Numerical experiments highlighting the performance of the proposed method are presented. Moreover, the competitiveness of the p-version DGFEM employing a 𝒫p-basis in comparison to the conforming p-version finite element method on tensor-product elements is studied numerically for a simple test problem.

scholarly journals Symmetry and reduction in collectives: low-dimensional cyclic pursuit

2016 ◽  
Vol 472 (2194) ◽  
pp. 20160465 ◽  
Author(s):  
Kevin S. Galloway ◽  
Eric W. Justh ◽  
P. S. Krishnaprasad
Keyword(s):  
Closed Loop ◽  
Physical Space ◽  
Relative Equilibria ◽  
Two Dimensions ◽  
Three Dimensions ◽  
Control Schemes ◽  
Loop Dynamics ◽  
Cyclic Pursuit ◽  
Steering Law ◽  
Low Dimensional

We investigate low-dimensional examples of cyclic pursuit in a collective, wherein each agent employs a constant bearing (CB) steering law relative to exactly one other agent. For the case of three agents in the plane, we characterize relative equilibria and pure shape equilibria of associated closed-loop dynamics. Re-scaling time yields a reduction of phase space to two dimensions and effective tools for stability analysis. Study of bifurcation of a family of collinear equilibria dependent on a single CB control parameter reveals the presence of a rich collection of trajectories that are periodic in shape and undergo precession in physical space. For collectives in three dimensions, with an appropriate notion of CB pursuit strategy and corresponding steering law, the two-agent case proves to be explicitly integrable. These results suggest control schemes for small teams of mobile robotic agents engaged in area coverage tasks such as search and rescue, and raise interesting possibilities for behaviour in biological contexts.

A NEW INTERIOR PENALTY DISCONTINUOUS GALERKIN METHOD FOR THE REISSNER–MINDLIN MODEL

2010 ◽  
Vol 20 (08) ◽  
pp. 1343-1361 ◽  
Author(s):  
PAULO R. BÖSING ◽  
ALEXANDRE L. MADUREIRA ◽  
IGOR MOZOLEVSKI
Keyword(s):  
Finite Element ◽  
Discontinuous Galerkin ◽  
A Priori ◽  
Mindlin Plate ◽  
Interior Penalty ◽  
Galerkin Finite Element ◽  
Discontinuous Galerkin Finite Element ◽  
Integration Techniques ◽  
Galerkin Finite Element Methods ◽  
Half Thickness

We introduce an interior penalty discontinuous Galerkin finite element method for the Reissner–Mindlin plate model that, as the plate's half-thickness ϵ tends to zero, recovers a hp interior penalty discontinuous Galerkin finite element methods for biharmonic equation. Our method does not introduce shear as an extra unknown, and does not need reduced integration techniques. We develop the a priori error analysis of these methods and prove error bounds that are optimal in h and uniform in ϵ. Numerical tests, that confirm our predictions, are provided.

MIXED hp-DISCONTINUOUS GALERKIN FEM FOR LINEAR ELASTICITY AND STOKES FLOW IN THREE DIMENSIONS

2012 ◽  
Vol 22 (08) ◽  
pp. 1250016 ◽  
Author(s):  
THOMAS P. WIHLER ◽  
MARCEL WIRZ
Keyword(s):  
Discontinuous Galerkin ◽  
Linear Elasticity ◽  
Degrees Of Freedom ◽  
Poisson Ratio ◽  
A Priori ◽  
Three Dimensions ◽  
Singular Solutions ◽  
Hexahedral Elements ◽  
Discontinuous Galerkin Fem ◽  
Stability Results

We consider mixed hp-discontinuous Galerkin FEM for linear elasticity in polyhedral domains Ω ⊂ ℝ3. In order to resolve possible corner, edge, and corner–edge singularities, anisotropic axiparallel geometric edge meshes consisting of hexahedral elements are applied. We show inf–sup stability results on both the continuous and the discrete level which are robust with respect to the Poisson ratio as it tends to the incompressible limit of ½. Furthermore, in the subsequent a priori error analysis we derive a quasi-optimality result, including the case of singular solutions. In addition, under certain realistic assumptions (for analytic data) on the regularity of the exact solution, we prove that the proposed DG schemes converge at an exponential rate in terms of the fifth root of the number of degrees of freedom.

On meshfree numerical differentiation

2018 ◽  
Vol 16 (05) ◽  
pp. 717-739
Author(s):  
Leevan Ling ◽  
Qi Ye
Keyword(s):  
A Priori ◽  
Noisy Data ◽  
Numerical Differentiation ◽  
Two Dimensions ◽  
Three Dimensions ◽  
Positive Definite Kernels ◽  
Approximation Quality ◽  
Symmetric Positive Definite ◽  
Optimal Kernel

We combine techniques in meshfree methods and Gaussian process regressions to construct kernel-based estimators for numerical derivatives from noisy data. Specially, we construct meshfree estimators from normal random variables, which are defined by kernel-based probability measures induced from symmetric positive definite kernels, to reconstruct the unknown partial derivatives from scattered noisy data. Our developed theories give rise to Tikhonov regularization methods with a priori parameter, but the shape parameters of the kernels remain tunable. For that, we propose an error measure that is computable without the exact values of the derivative. This allows users to obtain a quasi-optimal kernel-based estimator by comparing the approximation quality of kernel-based estimators. Numerical examples in two dimensions and three dimensions are included to demonstrate the convergence behavior and effectiveness of the proposed numerical differentiation scheme.

Hilbert maps: Scalable continuous occupancy mapping with stochastic gradient descent

2016 ◽  
Vol 35 (14) ◽  
pp. 1717-1730 ◽  
Author(s):  
Fabio Ramos ◽  
Lionel Ott
Keyword(s):  
Probability Distributions ◽  
A Priori ◽  
Two Dimensions ◽  
Stochastic Gradient ◽  
Three Dimensions ◽  
Stochastic Gradient Descent ◽  
Adaptive Capabilities ◽  
Localization And Mapping ◽  
Before And After ◽  
Benchmark Datasets

The vast amount of data robots can capture today motivates the development of fast and scalable statistical tools to model the space the robot operates in. We devise a new technique for environment representation through continuous occupancy mapping that improves on the popular occupancy grip maps in two fundamental aspects: (1) it does not assume an a priori discrimination of the world into grid cells and therefore can provide maps at an arbitrary resolution; (2) it captures spatial relationships between measurements naturally, thus being more robust to outliers and possessing better generalization performance. The technique, named Hilbert maps, is based on the computation of fast kernel approximations that project the data in a Hilbert space where a logistic regression classifier is learnt. We show that this approach allows for efficient stochastic gradient optimization where each measurement is only processed once during learning in an online manner. We present results with three types of approximations: random Fourier; Nyström; and a novel sparse projection. We also extend the approach to accept probability distributions as inputs, for example, due to uncertainty over the position of laser scans due to sensor or localization errors. In this extended version, experiments were conducted in two dimensions and three dimensions, using popular benchmark datasets. Furthermore, an analysis of the adaptive capabilities of the technique to handle large changes in the data, such as trajectory update before and after loop closure during simultaneous localization and mapping, is also included.

scholarly journals Anisotropic polygonal and polyhedral discretizations in finite element analysis

2019 ◽  
Vol 53 (2) ◽  
pp. 475-501 ◽  
Author(s):  
Steffen Weißer
Keyword(s):  
Finite Element Analysis ◽  
Adaptive Mesh Refinement ◽  
A Priori ◽  
Adaptive Mesh ◽  
Linear Mapping ◽  
Three Dimensions ◽  
Element Analysis ◽  
Reference Element ◽  
Polyhedral Meshes ◽  
Interpolation Operators

Interpolation and quasi-interpolation operators of Clément- and Scott-Zhang-type are analyzed on anisotropic polygonal and polyhedral meshes. Since no reference element is available, an appropriate linear mapping to a reference configuration plays a crucial role. A priori error estimates are derived respecting the anisotropy of the discretization. Finally, the found estimates are employed to propose an adaptive mesh refinement based on bisection which leads to highly anisotropic and adapted discretizations with general element shapes in two- and three-dimensions.

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