Continuum Methodology Research Articles

Generalized continuum methodologies in acoustics: Selected case studies

Three problems from acoustics theory are considered using both analytical and numerical methods. Each problem involves the use of a particular generalized continuum (GC) formulation. In the first, mechanical and thermal plane waves in gases wherein the specific internal energy is described by the modified internal energy model are analyzed; in the second, vertically-propagating acceleration waves in a dusty, constant lapse rate, atmosphere are investigated using the Laplace transform; and, in the third, dispersed shocks in gases wherein viscous stresses are described by the Stokesian fluid relation are examined under the traveling wave assumption. The presentation concludes with a listing of possible follow-on studies involving GC-based model systems.

A New Three-Dimensional Numerical Model Based on the Equivalent Continuum Method to Simulate Hydraulic Fracture Propagation in an Underground Coal Mine

Hydraulic fracturing has been proven to be the most efficient way to improve the permeability of coal seams. In this work, the hydraulic fracture propagation in an underground coal mine was numerically investigated. A new numerical approach was developed based on the equivalent continuum methodology to model the hydro-mechanical behavior of multiple fractures in three dimensions. It was solved using a hybrid combination of the embedded element method (EEM) and the finite volume method (FVM) using an iterative coupling schema. The FVM was employed to calculate the pressure field, while the EEM was used to track the displacement discontinuity caused by fractures. A fracture constitutive model was implemented to describe the aperture variation, shear slippage, and shear dilation for both contact and open fractures, as well as for contact and open criteria. To verify the developed model, two benchmark examples were presented. Then, the developed model was used to numerically investigate hydraulic fracture propagation in Datong underground coal mine in Songzao in Chongqing. According to the numerical study, it was found that (1) a fracture network created by a hydraulic fracturing operation in a coal mine is more complex than the ideal cross-cutting-shape, H-shape, T-shape, and Z-shape patterns; (2) the orientation of the minimum principal stress controlled the main propagation direction; (3) the complexity of the fracture pattern is controlled by the geological structure, the in situ stress and the injection rate.

Aggregation of photosensitizers: the role of dispersion and solvation on dimer formation energetics

A detailed account for the dimer formation energetics is computed for two commonly used photosensitizers: metal-free phthalocyanine (Pc) and methylene blue (MB), a cationic phenothiazine. Gas-phase calculations were performed using the density functional theory augmented with dispersion corrections using both the dispersion-correcting atom-centred potentials and the D2/D3 methods. SCS-MP2/CBS results were also provided for comparison. Hydration free energies were determined using continuum methodologies: namely the conductor-like polarizable continuum model, the solvation model density and a reparametrized version of the polarizable continuum model. For methylene blue, the Langevin dipole model was also used. Our results showed that the nature of aggregation is remarkably different for these two molecules, being dispersion driven for Pc while solvent driven for MB. For water and ethanol our results are in good agreement with the experimental data available for MB. We also provide quantitative estimates to the popular explanation put out by Mukerjee and Ghosh more than 40 years ago that aggregation arises from solvent entropic and inter-species enthalpic contributions.

Concurrent atomistic and continuum simulation of bi-crystal strontium titanate with tilt grain boundary.

In this paper, we present the development of a concurrent atomistic-continuum (CAC) methodology for simulation of the grain boundary (GB) structures and their interaction with other defects in ionic materials. Simulation results show that the CAC simulation allows a smooth passage of cracks through the atomistic-continuum interface without the need for additional constitutive rules or special numerical treatment; both the atomic-scale structures and the energies of the four different [001] tilt GBs in bi-crystal strontium titanate obtained by CAC compare well with those obtained by existing experiments and density function theory calculations. Although 98.4% of the degrees of freedom of the simulated atomistic system have been eliminated in a coarsely meshed finite-element region, the CAC results, including the stress-strain responses, the GB-crack interaction mechanisms and the effect of the interaction on the fracture strength, are comparable with that of all-atom molecular dynamics simulation results. In addition, CAC simulation results show that the GB-crack interaction has a significant effect on the fracture behaviour of bi-crystal strontium titanate; not only the misorientation angle but also the atomic-level details of the GB structure influence the effect of the GB on impeding crack propagation.

Concurrent atomistic and continuum simulation of strontium titanate

This paper presents a concurrent atomistic–continuum methodology (CAC) to simulate the dynamic processes of dislocation nucleation and migration as well as crack initiation and propagation in complex crystals. The accuracy and efficiency of the method is tested with respect to the molecular dynamics (MD) method through simulations of the dynamic fracture processes in strontium titanate under a combination of tension and shear loading and the dislocation behavior under nanoindentation. CAC simulation results demonstrated a smooth passage of cracks and dislocations through the atomistic–continuum interface without the need for additional constitutive rules or special numerical treatment. Although some accuracy is lost in CAC simulations as a consequence of a 98.4% reduction in the degrees of freedom, all the CAC results are qualitatively and quantitatively comparable with MD results. The stacking fault width and nanoindentation hardness measured in the CAC simulations agrees well with existing experimental data. Criteria for cleavage and slip in ionic materials are verified. The need to include the internal degrees of freedom of atoms in concurrent atomistic–continuum methods for polyatomic crystalline materials is confirmed.

Coarse-grained atomistic simulations of dislocations in Al, Ni and Cu crystals

This paper presents the application of a recently developed concurrent atomistic continuum (CAC) methodology in coarse-grained (CG) atomistic simulations of dislocation nucleation and migration in face-centered cubic (fcc) Al, Ni and Cu crystals, using an EAM force field and an adaptive mesh refinement strategy. The CAC method is based on recently developed atomistic field theory (AFT) that frames the problem in terms of a continuum field representation of lattice and sublattice atomic arrangements. Four sets of CG models with different finite element mesh refinement are constructed to test the accuracy and efficiency of the CG method relative to full molecular dynamics (MD). Simulation results show that the CG method is able to reproduce key phenomena of dislocation dynamics in initially defect free fcc crystals, including strain bursts caused by dislocation nucleation and migration, the structure of leading and trailing partial dislocations separated by equilibrium stacking faults in Al, formation of intrinsic stacking fault ribbons in Ni and Cu, and 3D migration of curved dislocations, all comparable with results of MD simulations. CG simulations have also revealed that the yield strength of Al depends on the thickness of the specimens as well as the operative deformation mechanisms. The effects of mesh size and adaptive mesh refinement on CG simulation results are analyzed and discussed.

A concurrent scheme for passing dislocations from atomistic to continuum domains

This paper presents a concurrent atomistic–continuum (CAC) methodology for three-dimensional dynamic simulation of dislocation nucleation, migration and interaction. The method is based on a new continuum field formulation of balance laws with relevant atomistic information (the arrangements and interactions of atoms) considered. In this work, we show that the new CAC method allows the smooth passage of dislocations through sharp interfaces between the atomistic and the coarse-grained finite element domains without unphysical reflection of dislocations or the need for heuristic rules; meanwhile, complex dislocation phenomena such as dislocation nucleation, dynamic strain bursts associated with nucleation and migration avalanches, formations of Lomer–Cottrell locks, dislocation–rigid boundary interactions, formation of intrinsic and extrinsic stacking faults, deformation twinning, and curved dislocation loops can be reproduced by the CAC method. All of the CAC simulations are directly compared with the corresponding atomic-level molecular dynamics (MD) simulations. The efficiency, accuracy and potential applications of the method are discussed along with necessary additional development of criteria for coarse graining.

Theory of flow distribution in manifolds

Flows in manifolds are of great importance in quite diverse fields of science and technology, including fuel cells, spargers, solar collectors, microchannels, porous infiltration and irrigation. Theory of flow distribution and pressure drop is vital to predict process performance and efficiency of manifold systems. In this paper, we examined research and development of theoretical models and methodology of solutions in flow in manifolds and highlight remarkable advances in the past fifty years. The main existing models and solution methods were unified further to one theoretical framework, including Bernoulli theory and momentum theory, and discrete and continuum methodologies. The generalised model was applicable to not only designs of continuum manifolds but also those of discrete manifolds with constant or varying factors. The procedure of design calculation is in reality straightforward without requirements of iteration, successive approximation and computer programme. The theoretical model provides easy-to-use design guidance to investigate the interactions among structures, operating conditions and manufacturing tolerance under a wide variety of combination of flow conditions and geometries through three general characteristic parameters ( E, M and ζ) and to minimize the impact on manifold operability.

Electrostatics in Proteins and Protein–Ligand Complexes

Accurate computational methods for predicting electrostatic energies are of major importance for our understanding of protein energetics in general for computer-aided drug design as well as for the design of novel biocatalysts and protein therapeutics. Electrostatic energies are of particular importance in such applications as virtual screening, drug design and protein-protein docking due to the high charge density of protein ligands and small-molecule drugs, and the frequent protonation state changes observed when drugs bind to their protein targets. Therefore, the development of a reliable and fast algorithm for the evaluation of electrostatic free energies, as an important contributor to the overall protein energy function, has been the focus for many scientists over the past three decades. In this review we describe the current state-of-the-art in modeling electrostatic effects in proteins and protein-ligand complexes. We focus mainly on the merits and drawbacks of the continuum methodology, and speculate on future directions in refining algorithms for calculating electrostatic energies in proteins using experimental data.

Preferential solvation of spherical ions in binary DMSO/benzene mixtures

We consider a new qualitative approach for treating theoretically the solvation of single-atomic ionic solutes in binary mixtures of polar and nonpolar aprotic solvents. It is based on the implicit continuum electrostatic model of the solvent mixture involving distance-dependent dielectric permittivity epsilon(R) (where R is the distance from the ion) and local concentrations C(1)(R) and C(2)(R) of the solvent ingredients. For a given R, the condition for local thermodynamic equilibrium provides the transcendental equation for explicitly establishing the permittivity and concentration profiles. Computations performed with real Cl(-) and model Cl(+) ions as solutes in benzene/DMSO mixtures are compared with the molecular dynamics simulations of the same systems. A significant discrepancy of molecular and continuum results is revealed for the concentration profiles in the close vicinity of the ion boundary, although the general trends are similar. The continuum methodology cannot account for the formation of rigid solvent structures around ions, which is most significant for the case of Cl(+). Such defect, however, proves to become of less importance in calculations of the solvation free energy, which are quite satisfactory for Cl(-) ion. Free energy calculations for Cl(+) are less successful in the range of low DMSO concentration.

Hybrid Usage of Computational Tools in Drug Synthesis

We describe several computational methodologies used in aiding the chemical synthesis of drugs. We first summarize quantum mechanical approaches that weigh thermodynamical and kinetic factors in selecting the possible pathways during synthesis. The two major problems encountered in computational approaches are the efficient sampling of the conformational space and the incorporation of solvent effect into the system of interest. Thus, conformational search methodologies of small to medium sized molecules, with emphasis on cyclic molecules, are reviewed. Also, the analysis of the solvent effect on the synthesis of drug molecules and yield, using continuum methodologies as well as molecular dynamics, is discussed. How results from these studies are in turn fed back into detailed quantum mechanical calculations with supermolecules of solvent and reaction site are outlined. It is shown that the usage of several computational techniques hand-in-hand provides a plethora of information that may be utilized during the actual synthesis of drug molecules. Keywords: Conformational analysis, molecular dynamics, quantum chemical computations, rational drug design, solvent effect, macrolide antibiotics

Polymer Matrix Effects on the Nonlinear Optical Response of Incorporated Chromophore: New Analytical Models

A new approach aimed at the modeling of the nonlinear optical (NLO) response of a dipole chromophore incorporated into a locally anisotropic, deformable, polarizable polymer matrix is suggested. The general continuum methodology is used with a specific cavity ansatz being employed; the cavity is chosen to be conformal to the characteristic ellipsoid of the generalized permittivity tensor of the polymer medium. The suggested ansatz allows the electrostatic boundary value problem to be solved analytically, and the local field experienced by the chromophore in the polymer electret to be found. Four analytically solvable models, which correspond to two singular and two nonsingular models, are considered; in two of them the chromophore is characterized by singular dipole and quadrupole moments; the other two use the approximation of the electric moment of the chromophore with that of the polarized ellipsoid. The relation between the macroscopic polymer properties and the microscopic characteristics of the NLO chromophore is established. The compliance of the obtained formulas for the local field with those of the classical Onsager approach is analyzed, and their specific features are considered.

On a general concept for the analysis of crack growth and material damage

Abstract For finite strain dynamics a variational model of crack evolution is formulated within the generalized oriented continuum methodology. In this approach position- and direction-dependent deformation and strain measures are used to describe the (macro)motion of the body with defects, which may evolve relative to the moving body. The inelastic behaviour of continua with evolving defects is represented by phenomenological equations including the transversal crack evolution. A strain-induced crack propagation criterion is defined by the difference between the strain energy release rate and the rate of the surface energy of the crack. A possible nucleation of microcracks in terms of the average drag coefficient of the crack configuration is proposed. Based on the crack growth criterion presented in this paper, the kinking of cracks is investigated using variational concepts. A constitutive damage model of Kachanov's type accounting for the crack density is derived in terms of the free energy functional and a dissipation potential.

Quantum modelling of reactions in solution: An overview of the dielectric continuum methodology

In this review, we report on the status and perspectives for theoretical modelling of solvent effects on reactions in solution using quantum-mechanical methods coupled with the dielectric continuum solvation methodology.The simplicity of the dielectric continuum approach allows inclusion of solvent effects directly into the solute Hamiltonian. Thus, it provides a practical tool for optimizing transition state structures and following reaction paths of reactions in solution in a rather routine manner as for gas-phase systems. Several different types of reaction are examined in more details to illustrate the advantages and disadvantages, accuracy and applicability of the continuum models employed.

Dynamic Analysis of Tall Building Using Reduced-Order Continuum Model

On Oct. 17, 1989, a tall, steel-framed, San Francisco office building with 47 above-grade stories was excited by the Loma Prieta earthquake. The building response was recorded by accelerometers maintained by the California Division of Mines and Geology Strong Motion Instrumentation Program. From the records, important characteristics of the building response can be identified. In this paper, a computationally efficient, approximate dynamic analysis of the building using a reduced-order continuum model is conducted. By making use of the repetitive nature of the lattice framing system, finite-element models generated using continuum methodology have significantly fewer degrees of freedom than models generated using classical discrete finite-element techniques. These reduced-order models can therefore lead to significant savings in the computational effort and cost typically associated with dynamic analyses of large structures. The vibrational characteristics of the continuum model, as well as the results of the dynamic analysis, are compared to the building’s recorded response. The continuum model is accurate in capturing the dominant periods of vibration as well as the time-history response and shows promise as a tool for preliminary design of large lattice structures.

Dynamic analysis of a ten-story reinforced concrete building using a continuum model

During the 1989 Loma Prieta earthquake, accelerometers maintained by the California Division of Mines and Geology Strong Motion Instrumentation Program (CSMIP) recorded the response of a 10-story, reinforced-concrete building located in San Jose, California. In this paper, a computationally efficient, approximate, dynamic analysis of the building is conducted utilizing a reduced-order continuum model. Continuum methodology makes use of the repetitive nature of lattice framing systems to generate finite element models having significantly fewer degrees of freedom than models generated using classical discrete finite element techniques. The vibrational characteristics of the continuum model, as well as the results of dynamic analyses, are compared to information gained from the recorded response of the building. The continuum model proves to be quite accurate, both in capturing the dominant periods of vibration of the structure and in predicting the time-history response. As a result, the method shows promise as a tool for use in the analysis and design of large lattice structures subjected to earthquake loads.

The selectivity of different external binding sites for quaternary ammonium ions in cloned potassium channels.

Tetraethylammonium (TEA) is thought to be the most effective quaternary ammonium (QA) ion blocker at the external site of K+ channels, and small changes to the TEA ion reduce its potency. To examine the properties of the external QA receptor, we applied a variety of QA ions to excised patches from human embryonic kidney cells or Xenopus oocytes transfected with the delayed rectifying K+ channels Kv 2.1 and Kv 3.1. In outside-out patches of Kv 3.1, the relative potencies were TEA > tetrapropylammonium (TPA) > tetrabutylammonium (TBA). In contrast to Kv 3.1, the relative potencies in Kv 2.1 were TBA > TEA > TPA. In Kv 3.1 and Kv 2.1, external tetrapentylammonium (TPeA) blocked K+ currents in a fast, reversible and, in contrast to TEA, time-dependent manner. The external binding of TPeA appeared to be voltage independent, unlike the effects of TPeA applied to inside-out patches. External n-alkyl-triethylammonium compounds (C8, C10 chain length) had a lower affinity than TEA in Kv 3.1, but a higher affinity than TEA in Kv 2.1. In Kv 3.1, the decrease in QA affinity was large when one or two methyl groups were substituted for ethyl groups in TEA, but minor when propyl groups replaced ethyl groups. Changes in the free energy of binding could be correlated to changes in the free energy of hydration of TEA derivatives calculated by continuum methodology. These results reveal a substantial hydrophobic component of external QA ion binding to Kv 2.1, and to a lesser degree to Kv 3.1, in addition to the generally accepted electrostatic interactions. The chain length of hydrophobic TEA derivatives affects the affinity for the hydrophobic binding site, whereas the hydropathy of QA ions determines the electrostatic interaction energy.

Analysis of Multiple‐Bay Frames Using Continuum Model

An approximate method that can be used to determine displacements and member forces of multiple-bay frames is presented. The method utilizes simple continuum models. Finite-element representations of building frameworks resulting from continuum methodology require significantly fewer degrees of freedom than classical discrete finite-element models, which individually model each of the beam and column elements. As a result, the use of continuum models to analyze structures can result in considerable savings in terms of computational effort. This makes the method attractive for use in the analysis of large frames, especially during the preliminary design stage. This paper reviews the continuum methodology and presents a method for computing discrete element forces from the continuum element displacements. In determining element forces, the effects of deep-beam-type cross-sectional warping are approximated. Examples including the analysis of fixed- and pinned-base single- and multiple-bay plane frames illustrate the accuracy of the continuum analyses procedure.

Strain interactions between dislocation arrays in epitaxial Si x Ge1-x /Si heterostructures

Lattice disregistry that exists in epitaxial overgrowths is often accommodated by interfacial dislocation arrays. The transition between strain accommodation by uniform interfacial shear and by interfacial dislocations is fairly sharp and thought to be controlled by energy minimization considerations. In this paper we demonstrate an extension of the Frank-van der Merwe approach by incorporating the continuum methodology of Eshelby to the analysis of strain interactions between arrays of interfacial dislocations in a bi-layered epitaxial film. Numerical examples are given for a Si/Si x Ge1-x /Si heterostructure. The importance of such an analysis to the study of defect propagation through strained layer superlattices is briefly discussed.

A continuum model for lattice structures with geometric and material nonlinearities

An equivalent continuum methodology offers an attractive alternative, or supplement, to classical discrete finite element procedures for the analysis of lattice structures. Effective continuum models can lead to reduced order models of lattice structures which are computationally very efficient. A number of continuum models have recently been proposed for analysis of lattice structures including one developed by the authors. The majority of these models have been concerned with linear analysis although some work has been done in the area of geometrical nonlinearities. To the authors' knowledge none of the existing continuum models have been extended to the analysis of lattices with general (i.e. both geometric and material) nonlinearities. The objective of the work reported herein was the development of a continuum model for general nonlinear analysis of lattice structures. The formulation of a continuum procedure for general nonlinear behavior is given. A continuum finite element is derived and a computational algorithm for nonlinear analysis is outlined. A number of applications of the continuum method for classical elasto-plastic material constitutive behavior are presented and compared to discrete finite element solutions. The examples illustrate the potential economy of the continuum finite element analysis vs classical discrete finite element analysis.