scholarly journals Variational Methods and Parallel Solvers in Chemo‐Mechanics

PAMM ◽  
2021 ◽  
Vol 20 (1) ◽  
Author(s):  
Björn Kiefer ◽  
Oliver Rheinbach ◽  
Stephan Roth ◽  
Friederike Röver
Keyword(s):  
Variational Methods ◽  
Parallel Solvers

Parallel Computing for Tire Simulations

2011 ◽  
Vol 39 (3) ◽  
pp. 193-209 ◽  
Author(s):  
H. Surendranath ◽  
M. Dunbar
Keyword(s):  
High Performance ◽  
Effective Means ◽  
Cost Effective ◽  
Turnaround Time ◽  
Careful Consideration ◽  
Rolling Contact ◽  
Element Analysis ◽  
Parallel Solvers ◽  
Design Changes ◽  
Highly Nonlinear

Abstract Over the last few decades, finite element analysis has become an integral part of the overall tire design process. Engineers need to perform a number of different simulations to evaluate new designs and study the effect of proposed design changes. However, tires pose formidable simulation challenges due to the presence of highly nonlinear rubber compounds, embedded reinforcements, complex tread geometries, rolling contact, and large deformations. Accurate simulation requires careful consideration of these factors, resulting in the extensive turnaround time, often times prolonging the design cycle. Therefore, it is extremely critical to explore means to reduce the turnaround time while producing reliable results. Compute clusters have recently become a cost effective means to perform high performance computing (HPC). Distributed memory parallel solvers designed to take advantage of compute clusters have become increasingly popular. In this paper, we examine the use of HPC for various tire simulations and demonstrate how it can significantly reduce simulation turnaround time. Abaqus/Standard is used for routine tire simulations like footprint and steady state rolling. Abaqus/Explicit is used for transient rolling and hydroplaning simulations. The run times and scaling data corresponding to models of various sizes and complexity are presented.

scholarly journals Multiple positive solutions for nonlocal elliptic problems involving the Hardy potential and concave–convex nonlinearities

2020 ◽  
pp. 2050008
Author(s):  
Shaya Shakerian
Keyword(s):  
Variational Methods ◽  
Bounded Domain ◽  
Positive Solutions ◽  
Variational Approach ◽  
Nehari Manifold ◽  
Multiple Positive Solutions ◽  
Hardy Potential ◽  
Smooth Bounded Domain ◽  
Existence And Multiplicity ◽  
The Nehari Manifold

In this paper, we study the existence and multiplicity of solutions for the following fractional problem involving the Hardy potential and concave–convex nonlinearities: [Formula: see text] where [Formula: see text] is a smooth bounded domain in [Formula: see text] containing [Formula: see text] in its interior, and [Formula: see text] with [Formula: see text] which may change sign in [Formula: see text]. We use the variational methods and the Nehari manifold decomposition to prove that this problem has at least two positive solutions for [Formula: see text] sufficiently small. The variational approach requires that [Formula: see text] [Formula: see text] [Formula: see text], and [Formula: see text], the latter being the best fractional Hardy constant on [Formula: see text].

Variational Methods for Machine Learning with Applications to Deep Networks

2021 ◽  
Author(s):  
Lucas Pinheiro Cinelli ◽  
Matheus Araújo Marins ◽  
Eduardo Antônio Barros da Silva ◽  
Sérgio Lima Netto
Keyword(s):  
Machine Learning ◽  
Variational Methods ◽  
Deep Networks

scholarly journals Multiplicity and concentration behaviour of solutions for a fractional Choquard equation with critical growth

2020 ◽  
Vol 10 (1) ◽  
pp. 732-774
Author(s):  
Zhipeng Yang ◽  
Fukun Zhao
Keyword(s):  
Small Parameter ◽  
Variational Methods ◽  
Critical Exponent ◽  
Laplace Operator ◽  
Sobolev Inequality ◽  
Fractional Laplacian ◽  
Singularly Perturbed ◽  
Choquard Equation ◽  
Fractional Choquard Equation ◽  
Fractional Laplace Operator

Abstract In this paper, we study the singularly perturbed fractional Choquard equation $$\begin{equation*}\varepsilon^{2s}(-{\it\Delta})^su+V(x)u=\varepsilon^{\mu-3}(\int\limits_{\mathbb{R}^3}\frac{|u(y)|^{2^*_{\mu,s}}+F(u(y))}{|x-y|^\mu}dy)(|u|^{2^*_{\mu,s}-2}u+\frac{1}{2^*_{\mu,s}}f(u)) \, \text{in}\, \mathbb{R}^3, \end{equation*}$$ where ε > 0 is a small parameter, (−△)s denotes the fractional Laplacian of order s ∈ (0, 1), 0 < μ < 3, $2_{\mu ,s}^{\star }=\frac{6-\mu }{3-2s}$is the critical exponent in the sense of Hardy-Littlewood-Sobolev inequality and fractional Laplace operator. F is the primitive of f which is a continuous subcritical term. Under a local condition imposed on the potential V, we investigate the relation between the number of positive solutions and the topology of the set where the potential attains its minimum values. In the proofs we apply variational methods, penalization techniques and Ljusternik-Schnirelmann theory.

Quantum mechanical algebraic variational methods for inelastic and reactive molecular collisions

1988 ◽  
Vol 92 (11) ◽  
pp. 3202-3216 ◽  
Author(s):  
David W. Schwenke ◽  
Kenneth Haug ◽  
Meishan Zhao ◽  
Donald G. Truhlar ◽  
Yan Sun ◽  
...  
Keyword(s):  
Variational Methods ◽  
Quantum Mechanical ◽  
Molecular Collisions
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