arXiv subjects

Grigor Nika

Publications and source records attributed to Grigor Nika.

6 recordsLinked to original sources

arXiv2024

Mathematics for energy systems: Methods, modeling strategies, and simulation

We offer an insight into our mathematical endeavors, which aim to advance the foundational understanding of energy systems in a broad context, encompassing facets such as charge transport, energy storage, markets, and collective behavior. Our working techniques include a combination of well-posed mathematical models (both deterministic and stochastic), mathematical analysis arguments (mostly concerned with model dimension reduction and averaging, periodic homogenization), and simulation tools (numerical approximation techniques, computational statistics, high-performance computing).

math.GM↗

arXiv2024

A Bound Preserving Energy Stable Scheme for a Nonlocal Cahn-Hilliard Equation

We present a finite-volume based numerical scheme for a nonlocal Cahn-Hilliard equation which combines ideas from recent numerical schemes for gradient flow equations and nonlocal Cahn-Hilliard equations. The equation of interest is a special case of a previously derived and studied system of equations which describes phase separation in ternary mixtures. We prove the scheme is both energy stable and respects the analytical bounds of the solution. Furthermore, we present numerical demonstrations of the theoretical results using both the Flory-Huggins (FH) and Ginzburg-Landau (GL) free-energy potentials.

math.NA↗

arXiv2024

Design and testing of 3D-printed micro-architectured polymer materials exhibiting a negative Poisson's ratio

This work proposes the complete design cycle for several auxetic materials where the cycle consists of three steps (i) the design of the micro-architecture, (ii) the manufacturing of the material and (iii) the testing of the material. We use topology optimization via a level-set method and asymptotic homogenization to obtain periodic micro-architectured materials with a prescribed effective elasticity tensor and Poisson's ratio. The space of admissible micro-architectural shapes that carries orthotropic material symmetry allows to attain shapes with an effective Poisson's ratio below -1. Moreover, the specimens were manufactured using a commercial stereolithography Ember printer and are mechanically tested. The observed displacement and strain fields during tensile testing obtained by digital image correlation match the predictions from the finite element simulations and demonstrate the efficiency of the design cycle.

physics.app-ph↗

arXiv2023

Effective medium theory for second-gradient elasticity with chirality

We derive effective, parsimonious models from a heterogeneous second-gradient nonlinear elastic material taking into account chiral scale-size effects. Our classification of the effective equations depends on the hierarchy of four characteristic lengths: The size of the heterogeneities $\ell$, the intrinsic lengths of the constituents $\ell_{\rm SG}$ and $\ell_{\rm chiral}$, and the overall characteristic length of the domain ${\rm L}$. Depending on the different scale interactions between $\ell_{\rm SG}$, $\ell_{\rm chiral}$, $\ell$, and ${\rm L}$ we obtain either an effective Cauchy continuum or an effective second-gradient continuum. The working technique combines scaling arguments with the periodic homogenization asymptotic procedure. Both the passage to the homogenization limit and the unveiling of the correctors' structure rely on a suitable use of the periodic unfolding and related operators.

math.AP↗

arXiv2019

Multiscale modeling of magnetorheological suspensions

We develop a multiscale approach to describe the behavior of a suspension of solid magnetizable particles in a viscous non-conducting fluid in the presence of an externally applied magnetic field. By upscaling the quasi-static Maxwell equations coupled with the Stokes' equations we are able to capture the magnetorheological effect. The model we obtain generalizes the one introduced by Neuringer & Rosensweig for quasistatic phenomena. We derive the macroscopic constitutive properties explicitly in terms of the solutions of local problems. The effective coefficients have a nonlinear dependence on the volume fraction when chain structures are present. The velocity profiles computed for some simple flows, exhibit an apparent yield stress and the flow profile resembles a Bingham fluid flow.

math.AP↗

arXiv2018

Design of multi-layer materials using inverse homogenization and a level set method

This work is concerned with the micro-architecture of multi-layer material that globally exhibits desired mechanical properties, for instance a negative apparent Poisson ratio. We use inverse homogenization, the level set method, and the shape derivative in the sense of Hadamard to identify material regions and track boundary changes within the context of the smoothed interface. The level set method and the shape derivative obtained in the smoothed interface context allows to capture, within the unit cell, the optimal micro-geometry. We test the algorithm by computing several multi-layer auxetic micro-structures. The multi-layer approach has the added benefit that contact during movement of adjacent "branches" of the micro-structure can be avoided in order to increase its capacity to withstand larger stresses.

math.OC↗