Anisotropic 2D mesh adaptation in hp-adaptive FEM

AbstractThe paper presents a grammar for anisotropic two-dimensional mesh adaptation in hp-adaptive Finite Element Method with rectangular elements. It occurs that a straightforward approach to modeling this process via grammar productions leads to potential deadlock in h-adaptation of the mesh. This fact is shown on a Petri net model of an exemplary adaptation. Therefore auxiliary productions are added to the grammar in order to ensure that any sequence of productions allowed by the grammar does not lead to a deadlock state. The fact that the enhanced grammar is deadlock-free is proven via a corresponding Petri net model. The proof has been performed by means of reachability graph construction and analysis. The paper is enhanced with numerical simulations of magnetolluric measurements where the deadlock problem occured

Petri Nets Modeling of Dead-End Refinement Problems in a 3D Anisotropic hp-Adaptive Finite Element Method

We consider two graph grammar based Petri nets models for anisotropic refinements of three dimensional hexahedral grids. The first one detects possible dead-end problems during the graph grammar based anisotropic refinements of the mesh. The second one employs an enhanced graph grammar model that is actually dead-end free. We apply the resulting algorithm to the simulation of resistivity logging measurements for estimating the location of underground oil and/or gas formations. The graph grammar based Petri net models allow to fix the self-adaptive mesh refinement algorithm and finish the adaptive computations with the required accuracy needed by the numerical solution

Petri Nets Modeling of Dead-End Refinement Problems in a 3D Anisotropic hp-Adaptive Finite Element Method

We consider two graph grammar based Petri nets models for anisotropic refinements of three dimensional hexahedral grids. The first one detects possible dead-end problems during the graph grammar based anisotropic refinements of the mesh. The second one employs an enhanced graph grammar model that is actually dead-end free. We apply the resulting algorithm to the simulation of resistivity logging measurements for estimating the location of underground oil and/or gas formations. The graph grammar based Petri net models allow to fix the self-adaptive mesh refinement algorithm and finish the adaptive computations with the required accuracy needed by the numerical solution

Output Error Control Using r-Adaptation

Peer Reviewedhttps://deepblue.lib.umich.edu/bitstream/2027.42/143062/1/6.2017-4111.pd

A summary of my twenty years of research according to Google Scholars

I am David Pardo, a researcher from Spain working mainly on numerical analysis applied to geophysics. I am 40 years old, and over a decade ago, I realized that my performance as a researcher was mainly evaluated based on a number called \h-index". This single number contains simultaneously information about the number of publications and received citations. However, dif- ferent h-indices associated to my name appeared in di erent webpages. A quick search allowed me to nd the most convenient (largest) h-index in my case. It corresponded to Google Scholars. In this work, I naively analyze a few curious facts I found about my Google Scholars and, at the same time, this manuscript serves as an experiment to see if it may serve to increase my Google Scholars h-index

Easy-to-implement hp-adaptivity for non-elliptic goal-oriented problems

The FEM has become a foundational numerical technique in computational mechanics and civil engineering since its inception by Courant in 1943 Courant1943. Originating from the Ritz method and variational calculus, the FEM was primarily employed to derive solutions for vibrational systems. A distinctive strength of the FEM is its capability to represent mathematical models through the weak variational formulation of PDE, facilitating computational feasibility even in intricate geometries. However, the search for accuracy often imposes a significant computational task. In the FEM, adaptive methods have emerged to balance the accuracy of solutions with computational costs. The $h$-adaptive FEM designs more efficient meshes by reducing the mesh size $h$ locally while keeping the polynomial order of approximation $p$ fixed (usually $p=1,2$). An alternative approach to the $h$-adaptive FEM is the $p$-adaptive FEM, which locally enriches the polynomial space $p$ while keeping the mesh size $h$ constant. By dynamically adapting $h$ and $p$, the $hp$-adaptive FEM achieves exponential convergence rates. Adaptivity is crucial for obtaining accurate solutions. However, the traditional focus on global norms, such as $L^2$ or $H^1$, might only sometimes serve the requirements of specific applications. In engineering, controlling errors in specific domains related to a particular QoI is often more critical than focusing on the overall solution. That motivated the development of GOA strategies. In this dissertation, we develop automatic GO $hp$-adaptive algorithms tailored for non-elliptic problems. These algorithms shine in terms of robustness and simplicity in their implementation, attributes that make them especially suitable for industrial applications. A key advantage of our methodologies is that they do not require computing reference solutions on globally refined grids. Nevertheless, our approach is limited to anisotropic $p$ and isotropic $h$ refinements. We conduct multiple tests to validate our algorithms. We probe the convergence behavior of our GO $h$- and $p$-adaptive algorithms using Helmholtz and convection-diffusion equations in one-dimensional scenarios. We test our GO $hp$-adaptive algorithms on Poisson, Helmholtz, and convection-diffusion equations in two dimensions. We use a Helmholtz-like scenario for three-dimensional cases to highlight the adaptability of our GO algorithms. We also create efficient ways to build large databases ideal for training DNN using $hp$ MAGO FEM. As a result, we efficiently generate large databases, possibly containing hundreds of thousands of synthetic datasets or measurements

An H-Adaptive Finite-Element Technique for Constructing 3D Wind Fields

An h-adaptive, mass-consistent finite-element model (FEM) has been developed for constructing 3D wind fields over irregular terrain utilizing sparse meteorological tower data. The element size in the computational domain is dynamically controlled by an a posteriori error estimator based on the L2 norm. In the h-adaptive FEM algorithm, large element sizes are typically associated with smooth flow regions and small errors; small element sizes are attributed to fast-changing flow regions and large errors. The adaptive procedure employed in this model uses mesh refinement–unrefinement to satisfy error criteria. Results are presented for wind fields using sparse data obtained from two regions within Nevada: 1) the Nevada Test Site, located approximately 65 mi (1 mi ~ 1.6 km) northwest of Las Vegas, and 2) the central region of Nevada, about 100 mi southeast of Reno