Computational Physics : Scipedia

Finite element methods for the non-linear mechanics of crystalline sheets and nanotubes
M. Arroyo, T. Belytschko

Int. J. Numer. Meth. Engng. (2004). Vol. 59 (3), pp. 419-456

Abstract
The formulation and finite element implementation of a finite deformation continuum theory for the mechanics of crystalline sheets is described. This theory generalizes standard crystal elasticity to curved monolayer lattices by means of the exponential Cauchy–Born rule. The constitutive model for a two‐dimensional continuum deforming in three dimensions (a surface) is written explicitly in terms of the underlying atomistic model. The resulting hyper‐elastic potential depends on the stretch and the curvature of the surface, as well as on internal elastic variables describing the rearrangements of the crystal within the unit cell. Coarse grained calculations of carbon nanotubes (CNTs) are performed by discretizing this continuum mechanics theory by finite elements. A smooth discrete representation of the surface is required, and subdivision finite elements, proposed for thin‐shell analysis, are used. A detailed set of numerical experiments, in which the continuum/finite element solutions are compared to the corresponding full atomistic calculations of CNTs, involving very large deformations and geometric instabilities, demonstrates the accuracy of the proposed approach. Simulations for large multi‐million systems illustrate the computational savings which can be achieved
Abstract
The formulation and finite element implementation of a finite deformation continuum theory for the mechanics of crystalline sheets is described. This theory generalizes standard [...]
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Modeling polycrystalline materials with elongated grains
I. Pérez, M. Farias, M. Castro, R. Roselló, C. Morfa, L. Medina, E. Oñate

Int J Numer Methods Eng. (2019). Vol. 118 (3), pp. 121-131

Abstract
A novel algorithm to reproduce the arrangement of grains in polycrystalline materials was recently published by the authors. In this original approach, a dense package of circles (or spheres) with the same distribution as the grains is generated to produce a set of Voronoi cells that are later modified to Laguerre cells representing the original structure. This algorithm was successfully applied to materials with somewhat equidimensional grains; however, it fails for long‐shaped grains. In this paper, modifications are provided in order to overcome these drawbacks. This is accomplished by moving each vertex of the Voronoi cells in such a way that the vertex should be equidistant from the particles with respect to the Euclidean distance. The algorithm is applied to packages of ellipses and spherocylinders in 2D. An example for a package of spheres is also provided to illustrate the application for a simple 3D case. The adherence between the generated packages and the corresponding tessellations is verified by means of the Jaccard coefficient (J). Several packages are generated randomly and the distribution of J coefficients is investigated. The obtained values satisfy the theoretical restraints and the quality of the proposed algorithm is statistically validated.
Abstract
A novel algorithm to reproduce the arrangement of grains in polycrystalline materials was recently published by the authors. In this original approach, a dense package of [...]
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Refficientlib: an efficient load-rebalanced adaptive mesh refinement algorithm for high-performance computational physics meshes
J. Baiges, C. Bayona

SIAM journal on scientific computing (2017). Vol. 39 (2), pp. C65-C95 (Preprint)

Abstract
In this paper we present a novel algorithm for adaptive mesh refinement in computational physics meshes in a distributed memory parallel setting. The proposed method is developed for nodally based parallel domain partitions where the nodes of the mesh belong to a single processor, whereas the elements can belong to multiple processors. Some of the main features of the algorithm presented in this paper are its capability of handling multiple types of elements in two and three dimensions (triangular, quadrilateral, tetrahedral, and hexahedral), the small amount of memory required per processor, and the parallel scalability up to thousands of processors. The presented algorithm is also capable of dealing with nonbalanced hierarchical refinement, where multirefinement level jumps are possible between neighbor elements. An algorithm for dealing with load rebalancing is also presented, which allows us to move the hierarchical data structure between processors so that load unbalancing is kept below an acceptable level at all times during the simulation. A particular feature of the proposed algorithm is that arbitrary renumbering algorithms can be used in the load rebalancing step, including both graph partitioning and space-filling renumbering algorithms. The presented algorithm is packed in the Fortran 2003 object oriented library \textttRefficientLib, whose interface calls which allow it to be used from any computational physics code are summarized. Finally, numerical experiments illustrating the performance and scalability of the algorithm are presented. No separate or additional fees are collected for access to or distribution of the work
Abstract
In this paper we present a novel algorithm for adaptive mesh refinement in computational physics meshes in a distributed memory parallel setting. The proposed method is developed [...]
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Finite crystal elasticity of carbon nanotubes based on the exponential Cauchy-Born rule
M. Arroyo, T. Belytschko

Physical review B: condensed matter and material physics (2004). Vol. 69 (11), pp. 115415-1, 115415-11

Abstract
A finite deformation continuum theory is derived from interatomic potentials for the analysis of the mechanics of carbon nanotubes. This nonlinear elastic theory is based on an extension of the Cauchy-Born rule called the exponential Cauchy-Born rule. The continuum object replacing the graphene sheet is a surface without thickness. The method systematically addresses both the characterization of the small strain elasticity of nanotubes and the simulation at large strains. Elastic moduli are explicitly expressed in terms of the functional form of the interatomic potential. The expression for the flexural stiffness of graphene sheets, which cannot be obtained from standard crystal elasticity, is derived. We also show that simulations with the continuum model combined with the finite element method agree very well with zero temperature atomistic calculations involving severe deformations.
Abstract
A finite deformation continuum theory is derived from interatomic potentials for the analysis of the mechanics of carbon nanotubes. This nonlinear elastic theory is based [...]
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Concurrent finite element simulation of quadrupolar and dipolar flow noise in low Mach number aeroacoustics
O. Guasch, A. Pont, J. Baiges, R. Codina

Computers and Fluids (2016). Vol. 133, pp. 129-139

Abstract
The computation of flow-induced noise at low Mach numbers usually relies on a two-step hybrid methodolgy. In the first step, an incompressible fluid dynamics simulation (CFD) is performed and an acoustic source term is derived from it. The latter becomes the inhomogeneous term for an acoustic wave equation, which is solved in the second step, often resorting to boundary integral formulations. In the presence of rigid bodies, Curle’s acoustic analogy is probably the most extended approach. It has been shown that Curle’s boundary dipolar noise contribution does in fact correspond to the diffraction of the quadrupolar aerodynamic noise generated by the flow past the rigid body. In this work, advantage is taken from this fact to propose an alternative computational methodology to get the individual quadrupolar and dipolar contributions to the total acoustic pressure. For any linear acoustic wave operator, the unknown acoustic pressure can be split into its incident and diffracted components and be computed simultaneously to the incompressible flow field, in a single finite element computational run. This circumvents the problem found in Curle’s analogy of needing the total pressure at the body’s boundary, which includes the acoustic pressure fluctuations. The latter cannot be obtained from an incompressible CFD simulation. The proposed unified strategy could be beneficial for a large variety problems such as those involving noise generated from duct terminations, or those related with the simulation of fricatives in numerical voice production, among many others.
Abstract
The computation of flow-induced noise at low Mach numbers usually relies on a two-step hybrid methodolgy. In the first step, an incompressible fluid dynamics simulation (CFD) [...]
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Driving mechanisms and streamwise homogeneity in molecular dynamics simulations of nanochannel flows
J. Principe, V. Bitrián

Physical review fluids (2018). (Preprint) Vol. 3 (1), pp. 1-16

Abstract
In molecular dynamics simulations, nanochannel flows are usually driven by a constant force, that aims to represent a pressure difference between inlet and outlet, and periodic boundary conditions are applied in the streamwise direction resulting in an homogeneous flow. The homogeneity hypothesis can be eliminated adding reservoirs at the inlet and outlet of the channel which permits us to predict streamwise variation of flow properties. It also opens the door to drive the flow by applying a pressure gradient instead of a constant force. We analyze the impact of these modeling modifications in the prediction of the flow properties, and we show when they make a difference with respect to the standard approach. It turns out that both assumptions are irrelevant when low pressure differences are considered, but important differences are observed at high pressure differences. They include the density and velocity variation along the channel (the mass flow rate is constant) but, more importantly, the temperature increase and slip length decrease. Because viscous heating is important at high shear rates, these modeling issues are also linked to the use of thermostating procedures. Specifically, selecting the region where the thermostat is applied has a critical influence on the results. Whereas in the traditional homogeneous model the choices are limited to the fluid and/or the wall, in the inhomogeneous cases the reservoirs are also available, which permits us to leave the region of interest, the channel, unperturbed.
Abstract
In molecular dynamics simulations, nanochannel flows are usually driven by a constant force, that aims to represent a pressure difference between inlet and outlet, and periodic [...]
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Size-dependent nonlinear elastic scaling of multiwalled carbon nanotubes
I. Arias, M. Arroyo

Physical review letters (2008). Vol. 100 (8), pp. 085503-1/085503-4

Abstract
We characterize through large-scale simulations the nonlinear elastic response of multi-walled carbon nanotubes (MWNCNTs) in torsion and bending. We identify a unified law consisting of two distinct power-law regimes in the energy-deformation relation. This law encapsulates the complex mechanics of rippling and is described in terms of elastic constants, a critical length-scale and an anharmonic energy-deformation exponent. The mechanical response of MWCNTs is found to be strongly size-dependent, in that the critical strain beyond which they behave nonlinearly scales as the inverse of their diameter. These predictions are consistent with available experimental observations.
Abstract
We characterize through large-scale simulations the nonlinear elastic response of multi-walled carbon nanotubes (MWNCNTs) in torsion and bending. We identify a unified law [...]
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Thermoelastic generation of ultrasound by line-focused laser irradiation
I. Arias, J. Achenbach

Int. J. Solids and Structures (2003). Vol. 40 (25), pp. 6917-6935

Abstract
A two-dimensional theoretical model for the field generated in the thermoelastic regime by line-focused laser illumination of a homogeneous, isotropic, linearly elastic half-space is presented. The model accounts for the effects of thermal diffusion and optical penetration, as well as the finite width and duration of the laser source. The model is obtained by solving the thermoelastic problem in plane strain, rather than by integrating available solutions for the point-source, leading to a lower computational effort. The well-known dipole model follows from appropriate limits. However, it is shown that, by simple elasticity arguments, the strength of the dipole can be related a-priori to the heat input and certain material properties. The strength is found to be smaller than that of the dipoles equivalent to a buried source due to the effect of the free surface. This fact has been overlooked by some previous researchers. Excellent quantitative agreement with experimental observations provides validation for the model. Some representative results are presented to illustrate the generated field and provide insight into the relevance of the different mechanisms taken into account in the model.
Abstract
A two-dimensional theoretical model for the field generated in the thermoelastic regime by line-focused laser illumination of a homogeneous, isotropic, linearly elastic half-space [...]
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Force transduction and lipid binding in MscL: a continuum-molecular approach
J. Vanegas, M. Arroyo

PLoS one (2014). Vol. 9 (12), pp. e113947 (preprint)

Abstract
The bacterial mechanosensitive channel MscL, a small protein mainly activated by membrane tension, is a central model system to study the transduction of mechanical stimuli into chemical signals. Mutagenic studies suggest that MscL gating strongly depends on both intra-protein and interfacial lipid-protein interactions. However, there is a gap between this detailed chemical information and current mechanical models of MscL gating. Here, we investigate the MscL bilayer-protein interface through molecular dynamics simulations, and take a combined continuum-molecular approach to connect chemistry and mechanics. We quantify the effect of membrane tension on the forces acting on the surface of the channel, and identify interactions that may be critical in the force transduction between the membrane and MscL. We find that the local stress distribution on the protein surface is largely asymmetric, particularly under tension, with the cytoplasmic side showing significantly larger and more localized forces, which pull the protein radially outward. The molecular interactions that mediate this behavior arise from hydrogen bonds between the electronegative oxygens in the lipid headgroup and a cluster of positively charged lysine residues on the amphipathic S1 domain and the C-terminal end of the second trans-membrane helix. We take advantage of this strong interaction (estimated to be 10-13 kT per lipid) to actuate the channel (by applying forces on protein-bound lipids) and explore its sensitivity to the pulling magnitude and direction. We conclude by highlighting the simple motif that confers MscL with strong anchoring to the bilayer, and its presence in various integral membrane proteins including the human mechanosensitive channel K2P1 and bovine rhodopsin.
Abstract
The bacterial mechanosensitive channel MscL, a small protein mainly activated by membrane tension, is a central model system to study the transduction of mechanical stimuli [...]
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Geometric derivation of the microscopic stress: a covariant central force decomposition
A. Torres-Sánchez, J. Vanegas, M. Arroyo

Journal of the Mechanics and Physics of Solids (2016). Vol. 93, pp. 224-239 (preprint)

Abstract
We revisit the derivation of the microscopic stress, linking the statistical mechanics of particle systems and continuum mechanics. The starting point in our geometric derivation is the Doyle-Ericksen formula, which states that the Cauchy stress tensor is the derivative of the free-energy with respect to the ambient metric tensor and which follows from a covariance argument. Thus, our approach to define the microscopic stress tensor does not rely on the statement of balance of linear momentum as in the classical Irving-Kirkwood-Noll approach. Nevertheless, the resulting stress tensor satisfies balance of linear and angular momentum. Furthermore, our approach removes the ambiguity in the definition of the microscopic stress in the presence of multibody interactions by naturally suggesting a canonical and physically motivated force decomposition into pairwise terms, a key ingredient in this theory. As a result, our approach provides objective expressions to compute a microscopic stress for a system in equilibrium and for force-fields expanded into multibody interactions of arbitrarily high order. We illustrate the proposed methodology with molecular dynamics simulations of a fibrous protein using a force-field involving up to 5-body interactions.
Abstract
We revisit the derivation of the microscopic stress, linking the statistical mechanics of particle systems and continuum mechanics. The starting point in our geometric derivation [...]
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Examining the mechanical equilibrium of microscopic stresses in molecular simulations
A. Torres-Sánchez, J. Vanegas, M. Arroyo

Physical review letters (2015). Vol. 114 (25), pp. 258102-1/258102-5 (Preprint)

Abstract
The microscopic stress field provides a unique connection between atomistic simulations and mechanics at the nanoscale. However, its definition remains ambiguous. Rather than a mere theoretical preoccupation, we show that this fact acutely manifests itself in local stress calculations of defective graphene, lipid bilayers, and fibrous proteins. We find that popular definitions of the microscopic stress violate the continuum statements of mechanical equilibrium, and we propose an unambiguous and physically sound definition.
Abstract
The microscopic stress field provides a unique connection between atomistic simulations and mechanics at the nanoscale. However, its definition remains ambiguous. Rather than [...]
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Three-dimensional simulation of crack propagation in ferroelectric polycrystals: Effect of combined toughening mechanisms
A. Abdollahi, I. Arias

Acta Materialia (2014). Vol. 65, pp. 106-117 (Preprint)

Abstract
We simulate the fracture processes of ferroelectric polycrystals in three dimensions using a phase-field model. In this model, the grain boundaries, cracks and ferroelectric domain walls are represented in a diffuse way by three phase-fields. We thereby avoid the difficulty of tracking the interfaces in three dimensions. The resulting model can capture complex interactions between the crack and the polycrystalline and ferroelectric domain microstructures. The simulation results show the effect of the microstructures on the fracture response of the material. Crack deflection, crack bridging, crack branching and ferroelastic domain switching are observed to act as the main fracture toughening mechanisms in ferroelectric polycrystals. Our fully 3-D simulations illustrate how the combination of these mechanisms enhances the fracture toughness of the material, and pave the way for further systematic studies, including fracture homogenization.
Abstract
We simulate the fracture processes of ferroelectric polycrystals in three dimensions using a phase-field model. In this model, the grain boundaries, cracks and ferroelectric [...]
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On atomistic-to-continuum coupling by blending
S. Badia, M. Parks, P. Bochev, M. Gunzburger, R. Lehoucq

Multiscale modeling and simulation (2008). Vol. 7 (1), pp. 381-406

Abstract
A mathematical framework for the coupling of atomistic and continuum models by blending them over a subdomain subject to a constraint is developed. Using the framework, four classes of atomistic-to-continuum (AtC) blending methods are established, their consistency is studied, and their relative merits are discussed. In addition, the framework helps clarify the origin of ghost forces and formalizes the notion of a patch test. Numerical experiments with the AtC methods are used to illustrate the theoretical results.
Abstract
A mathematical framework for the coupling of atomistic and continuum models by blending them over a subdomain subject to a constraint is developed. Using the framework, [...]
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Optimization of a novel micro-opto-X-ray imaging lens
H. Ostadi, M. Arroyo, P. Prewett, K. Jiang, S. Huq

Microelestronic engineering (2008). Vol. 85 (5-6), pp. 1086-1088

Abstract
A mechanically deformable reflection transmission MOEMS system is capable of focusing X-rays to sub micron spots. This paper considers the geometry of the proposed deformed slotted microcantilever lens element under thermally derived strain, using finite element analysis. The work shows that an optimized MOEMS design using a slotted polyimide/gold thermal bimorph cantilever is capable of achieving ideal geometry with a larger than expected number of focusing slots – up to 111 at 2 μm width.
Abstract
A mechanically deformable reflection transmission MOEMS system is capable of focusing X-rays to sub micron spots. This paper considers the geometry of the proposed deformed [...]
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Nonlinear mechanical response and rippling of thick multi-walled carbon nanotubes
M. Arroyo, T. Belytschko

Physical review letters (2003). Vol. 91 (21), pp. 215505-1, 221505-4

Abstract
The measured drop of the effective bending stiffness of multiwalled carbon nanotubes (MWCNTs) with increasing diameter is investigated by a generalized local quasicontinuum method. The previous hypothesis that this reduction is due to a rippling mode is confirmed by the calculations. The observed ripples result from a complex three-dimensional deformation similar to theYoshimura buckling pattern. It is found that thick MWCNTs exhibit a well-defined nonlinear moment-curvature relation, even for small deformations, governed by the interplay of strain energy relaxation and intertube interactions. Rippling deformations are also predicted for MWCNTs subject to torsion, resulting in an effective torsional modulus much smaller than that predicted by linear elasticity.
Abstract
The measured drop of the effective bending stiffness of multiwalled carbon nanotubes (MWCNTs) with increasing diameter is investigated by a generalized local quasicontinuum [...]
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Rippling and a phase-transforming mesoscopic model for multiwalled carbon nanotubes
M. Arroyo, I. Arias

Journal of the Mechanics and Physics of Solids (2008). Vol. 56 (4), pp. 1124-1244

Abstract
We propose to model thick multiwalled carbon nanotubes as beams with non-convex curvature energy. Such models develop stressed phase mixtures composed of smoothly bent sections and rippled sections. This model is motivated by experimental observations and large-scale atomistic-based simulations. The model is analyzed, validated against large-scale simulations, and exercised in examples of interest. It is shown that modelling MWCNTs as linear elastic beams can result in poor approximations that overestimate the elastic restoring force considerably, particularly for thick tubes. In contrast, the proposed model produces very accurate predictions both of the restoring force and of the phase pattern. The size effect in the bending response of MWCNTs is also discussed.
Abstract
We propose to model thick multiwalled carbon nanotubes as beams with non-convex curvature energy. Such models develop stressed phase mixtures composed of smoothly bent sections [...]
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Modeling and enhanced sampling of molecular systems with smooth and nonlinear data-driven collective variables
B. Hashemian, D. Millán, M. Arroyo

Journal of chemical physics (2013). Vol. 139, pp. 214101/214101-12

Abstract
Collective variables (CVs) are low-dimensional representations of the state of a complex system, which help us rationalize molecular conformations and sample free energy landscapes with molecular dynamics simulations. Given their importance, there is need for systematic methods that effectively identify CVs for complex systems. In recent years, nonlinear manifold learning has shown its ability to automatically characterize molecular collective behavior. Unfortunately, these methods fail to provide a differentiable function mapping high-dimensional configurations to their low-dimensional representation, as required in enhanced sampling methods. We introduce a methodology that, starting from an ensemble representative of molecular flexibility, builds smooth and nonlinear data-driven collective variables (SandCV) from the output of nonlinear manifold learning algorithms. We demonstrate the method with a standard benchmark molecule, alanine dipeptide, and show how it can be non-intrusively combined with off-the-shelf enhanced sampling methods, here the adaptive biasing force method. We illustrate how enhanced sampling simulations with SandCV can explore regions that were poorly sampled in the original molecular ensemble. We further explore the transferability of SandCV from a simpler system, alanine dipeptide in vacuum, to a more complex system, alanine dipeptide in explicit water.
Abstract
Collective variables (CVs) are low-dimensional representations of the state of a complex system, which help us rationalize molecular conformations and sample free energy landscapes [...]
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Finite element approximation of nematic liquid crystal flows using a saddle-point structure
S. Badia, F. Guillén-González, J. Gutiérrez-Santacreu

Journal of Computational Physics (2011). Vol. 230 (4), pp. 1686-1706
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Simulation of high temperature superconductors and experimental validation
M. Olm, S. Badia, A. Martín

Computer Physics Communications (2019). Vol. 237, pp. 154-167

Abstract
In this work, we present a parallel, fully-distributed finite element numerical framework to simulate the low-frequency electromagnetic behaviour of superconducting devices, which efficiently exploits high performance computing platforms. We select the so-called H-formulation, which uses the magnetic field as a state variable. Nédélec elements (of arbitrary order) are required for an accurate approximation of the H-formulation for modelling electromagnetic fields along interfaces between regions with high contrast medium properties. An h-adaptive mesh refinement technique customized for Nédélec elements leads to a structured fine mesh in areas of interest whereas a smart coarsening is obtained in other regions. The composition of a tailored, robust, parallel nonlinear solver completes the exposition of the developed tools to tackle the problem. First, a comparison against experimental data is performed to show the availability of the finite element approximation to model the physical phenomena. Then, a selected state-of-the-art 3D benchmark is reproduced, focusing on the parallel performance of the algorithms. -
Abstract
In this work, we present a parallel, fully-distributed finite element numerical framework to simulate the low-frequency electromagnetic behaviour of superconducting [...]
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Thin-shell theory based analysis of radially pressurized multiwall carbon nanotubes
H. Shima, M. Arroyo, K. Iiboshi, M. Sato

Computational Materials Science (2012). Vol. 52 (1), pp. 90-94

Abstract
The radial deformation of multiwall carbon nanotubes (MWNTs) under hydrostatic pressure is investi gated within the continuum elastic approximation. A thin shell theory, with accurate elastic constants and interwall couplings, allows us to estimate the critical pressure above which the original circular cross section transforms into radially corrugated ones. The emphasis is placed on the rigorous formula tion of the van der Waals interaction between adjacent walls, which we analyze using two different approaches. Possible consequences of the radial corrugation in the physical properties of pressurized MWNTs are also discussed.
Abstract
The radial deformation of multiwall carbon nanotubes (MWNTs) under hydrostatic pressure is investi gated within the continuum elastic approximation. A thin shell theory, with [...]
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Finite element methods for the non-linear mechanics of crystalline sheets and nanotubes

Modeling polycrystalline materials with elongated grains

Refficientlib: an efficient load-rebalanced adaptive mesh refinement algorithm for high-performance computational physics meshes

Finite crystal elasticity of carbon nanotubes based on the exponential Cauchy-Born rule

Concurrent finite element simulation of quadrupolar and dipolar flow noise in low Mach number aeroacoustics

Driving mechanisms and streamwise homogeneity in molecular dynamics simulations of nanochannel flows

Size-dependent nonlinear elastic scaling of multiwalled carbon nanotubes

Thermoelastic generation of ultrasound by line-focused laser irradiation

Force transduction and lipid binding in MscL: a continuum-molecular approach

Geometric derivation of the microscopic stress: a covariant central force decomposition

Examining the mechanical equilibrium of microscopic stresses in molecular simulations

Three-dimensional simulation of crack propagation in ferroelectric polycrystals: Effect of combined toughening mechanisms

On atomistic-to-continuum coupling by blending

Optimization of a novel micro-opto-X-ray imaging lens

Nonlinear mechanical response and rippling of thick multi-walled carbon nanotubes

Rippling and a phase-transforming mesoscopic model for multiwalled carbon nanotubes

Modeling and enhanced sampling of molecular systems with smooth and nonlinear data-driven collective variables

Finite element approximation of nematic liquid crystal flows using a saddle-point structure

Simulation of high temperature superconductors and experimental validation

Thin-shell theory based analysis of radially pressurized multiwall carbon nanotubes

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