Fictitious Density Research Articles

Prediction of the Derivative Discontinuity in Density Functional Theory from an Electrostatic Description of the Exchange and Correlation Potential

We propose an approach to approximate the exchange and correlation (XC) term in density functional theory. The XC potential is considered as an electrostatic potential, generated by a fictitious XC density, which is in turn a functional of the electronic density. We apply the approach to develop a correction scheme that fixes the asymptotic behavior of any approximated XC potential for finite systems. Additionally, the correction procedure gives the value of the XC derivative discontinuity; therefore, it can directly predict the fundamental gap as a ground-state property.

Study on connections of the MFS, Trefftz method, indirect BIEM and invariant MFS in the three-dimensional Laplace problems containing spherical boundaries

Following the success of a study on the method of fundamental solutions using an image concept [13], we extend to solve the three-dimensional Laplace problems containing spherical boundaries by using the three approaches. The case of eccentric sphere for the Laplace problem is considered. The optimal locations for the source distribution to include the foci in the MFS are also examined by using the image concept in the 3D problems. Whether a free constant is required or not in the MFS is also studied. The error distribution is discussed after comparing with the analytical solution derived by using the bispherical coordinates. Besides, the relationship between the Trefftz bases and the singularity in the MFS for the three-dimensional Laplace problems is also addressed. It is found that one source of the MFS contains several interior and exterior Trefftz sets through a degenerate kernel. On the contrary, one single Trefftz base can be superimposed by some lumped sources in the MFS through an indirect BIEM. Based on this finding, the relationship between the fictitious boundary densities of the indirect BIEM and the singularity strength in the MFS can be constructed due to the fact that the MFS is a lumped version of an indirect BIEM.

Critical scaling near jamming transition for frictional granular particles

The critical rheology of sheared frictional granular materials near jamming transition is numerically investigated. It is confirmed that there exists a true critical density which characterizes the onset of the yield stress and two fictitious critical densities which characterize the scaling laws of rheological properties. We find the existence of a hysteresis loop between two of the critical densities for each friction coefficient. It is noteworthy that the critical scaling law for frictionless jamming transition seems to be still valid even for frictional jamming despite using fictitious critical density values.

Modeling Solvatochromic Shifts Using the Orbital-Free Embedding Potential at Statistically Mechanically Averaged Solvent Density

The correspondence between the exact embedding potential and the pair of the electron densities--that of the embedded molecule and that of its environment [Wesolowski and Warshel, J. Phys. Chem. 1993, 97, 8050]--is used to generate the average embedding potential and to subsequently calculate the solvatochromic shifts in a number of organic chromophores in solvents of various polarities. The averaged embedding potential is evaluated at a fictitious electron density of the solvent, which is obtained by means of "dressing up" with electrons the classical site distributions derived from the statistical-mechanical, 3D molecular theory of solvation (aka 3D-RISM method) [Kovalenko In Molecular Theory of Solvation; Hirata, Ed.; Understanding Chemical Reactivity; 2003, Vol 24], self-consistently coupled with the electronic structure of the solute. The proposed approach to modeling solvatochromic shifts can be situated between the implicit and explicit type of models for the solvent. Numerical examples are given for the lowest-lying n --> pi* and pi --> pi* excitations.

Multi‐actuated functionally graded piezoelectric micro‐tools design: A multiphysics topology optimization approach

AbstractMicro‐tools offer significant promise in a wide range of applications such as cell manipulation, micro‐surgery, and micro/nanotechnology processes. Such special micro‐tools consist of multi‐flexible structures actuated by two or more piezoceramic devices that must generate output displacements and forces at different specified points of the domain and at different directions. The micro‐tool structure acts as a mechanical transformer by amplifying and changing the direction of the piezoceramics output displacements. The design of these micro‐tools involves minimization of the coupling among movements generated by various piezoceramics. To obtain enhanced micro‐tool performance, the concept of multifunctional and functionally graded materials is extended by tailoring elastic and piezoelectric properties of the piezoceramics while simultaneously optimizing the multi‐flexible structural configuration using multiphysics topology optimization. The design process considers the influence of piezoceramic property gradation and also its polarization sign. The method is implemented considering continuum material distribution with special interpolation of fictitious densities in the design domain. As examples, designs of a single piezoactuator, an XY nano‐positioner actuated by two graded piezoceramics, and a micro‐gripper actuated by three graded piezoceramics are considered. The results show that material gradation plays an important role to improve actuator performance, which may also lead to optimal displacements and coupling ratios with reduced amount of piezoelectric material. The present examples are limited to two‐dimensional models because many of the applications for such micro‐tools are planar devices. Copyright © 2008 John Wiley & Sons, Ltd.

Elastic stress analysis of non-linear variable thickness rotating disks subjected to thermal load and having variable density along the radius

An analytical procedure for evaluation of elastic stresses and strains in non-linear variable thickness rotating disks, either solid or annular, subjected to thermal load, and having a fictitious density variation along the radius is presented. Thickness variation of disks is described by means of a power of linear function, which can be used to describe a fourfold infinity of actual disk profiles. The procedure is based on two independent integrals of the hypergeometric differential equation describing the displacement field; this theoretical procedure is just general and does not present limitations and drawbacks of the approaches as the one found in technical literature. General unpublished relations of stress state and displacement field in non-linear variable thickness disks subjected, under elastic conditions, to thermal gradient, and featuring a density variation along the radius are defined. Particular consideration is given to some industrial example of turbine rotors carrying hub and rim with buckets on periphery or radial blades on lateral surfaces. The analytical results obtained by using the new general relations perfectly match those obtained by FEA and overlap those concerning the special cases of tapered conical disks found in literature.

Elastic stress analysis of rotating converging conical disks subjected to thermal load and having variable density along the radius

An analytical procedure for evaluation of elastic stresses and strains in rotating conical disks, either solid or annular, subjected to thermal load, and having a fictitious density variation along the radius is presented. The procedure is based on two independent integrals of the hypergeometric differential equation describing the displacement field; this procedure is just as general as the one found in technical literature, but less intricate and more reliable. General unpublished relations of stress state and displacement field in conical disk subjected, under elastic conditions, to thermal gradient, and featuring a density variation along the radius are defined. Particular consideration is given to some industrial example of turbine rotors carrying hub and rim with buckets on periphery or radial blades on lateral surfaces. The analytical results obtained by using the new general relations perfectly match those obtained by FEA.

Optimum placement of piezoelectric material in piezoactuator design

Piezoelectric actuators offer significant promise in a wide range of applications. Thepiezoelectric actuators considered in this work essentially consist of a flexible structureactuated by piezoceramics that must generate output displacement and force at a certainspecified direction and point of the domain. The design of these piezoelectric actuators iscomplex, and a systematic design method such as topology optimization hasbeen successfully applied in recent years, with appropriate formulation of theoptimization problem to obtain optimized designs. However, in these previous designformulations, the position of the piezoceramic is usually kept fixed in the designdomain and only the flexible part of the structure is designed by distributing somenon-piezoelectric material (aluminum, for example). This imposes a constraint inthe position of the piezoelectric material in the optimization problem, limitingthe optimality of the solution. Thus, in this work, a formulation that allows thesimultaneous distribution of non-piezoelectric and piezoelectric material in the designdomain to achieve certain specified actuation movements will be presented. Theoptimization problem is posed as the simultaneous search for an optimal topology ofa flexible structure as well as the optimal position of the piezoceramic in thedesign domain and optimal rotation angles of piezoceramic material axes thatmaximize output displacements or output forces in a certain specified directionand point of the domain. The method is implemented based on the SIMP (‘solidisotropic material with penalization’) material model where fictitious densities areinterpolated in each finite element, providing a continuum material distributionin the domain. The examples presented are limited to two-dimensional models,since most of the applications for such piezoelectric actuators are planar devices.

Design of piezoelectric multi-actuated microtools using topology optimization

Microtools offer significant promise in a wide range of applications such as cellmanipulation, microsurgery, nanotechnology processes, and many other fields.The development of these microtools is still in the initial stages and it can bestrongly enhanced by using design tools. The microtools considered in this paperessentially consist of a multi-flexible structure actuated by two or more piezoceramicdevices such that when each piezoceramic is actuated, it generates an outputdisplacement and force in a specified point of the domain and direction. Themulti-flexible structure acts as a mechanical transform by amplifying and changing thedirection of the piezoceramic output displacements. Thus, the development ofmicrotools requires the design of actuated flexible structures that can perform complexmovements. In addition, when multiple piezoceramic devices are involved, couplingeffects in their movements become critical, especially the appearance of undesiredmovements, which makes the design task very complex. One way to avoid suchundesirable effects is the use of a systematic design method, such as topologyoptimization, with appropriate formulation of the optimization problem. Here, a topologyoptimization formulation is developed for the design of microtools actuated bymultiple piezoceramics that minimizes the effects of movement coupling. Thismethod is implemented based on the CAMD (continuous approximation of materialdistribution) approach where fictitious densities are interpolated in each finite element,providing a continuum material distribution in the domain. In addition, in previouspiezoelectric actuator topology optimization works the piezoceramics were excited bycharge, which is not realistic, even though it simplifies the sensitivity analysis.In this work, the piezoceramics are excited by voltage and the correspondingsensitivity analysis is presented. Different types of microtools required for variousapplications are designed to demonstrate the usefulness of the proposed method.Although the presented examples are limited to two-dimensional models, this isappropriate since most of the applications for such microtools are planar devices.

Optimization of traffic flow conditions by minimization of quadratic functionals

This paper deals with a scalar hydrodynamic model of traffic flow. In this model, the driver adapts his velocity to a fictitious density that he feels in nonequilibrium conditions, and he makes an effort to optimize the comfort and to minimize fuel consumption and gas emission. These optimization problems are here considered and analyzed by introducing and minimizing a quadratic functional. The analysis is developed on the basis of suitable simulations.

Optimization of traffic flow conditions by minimization of quadratic functionals

This paper deals with a scalar hydrodynamic model of traffic flow. In this model, the driver adapts his velocity to a fictitious density that he feels in nonequilibrium conditions, and he makes an effort to optimize the comfort and to minimize fuel consumption and gas emission. These optimization problems are here considered and analyzed by introducing and minimizing a quadratic functional. The analysis is developed on the basis of suitable simulations.

Unified formulation of radiation conditions for the wave equation

AbstractA family of radiation boundary conditions for the wave equation is derived by truncating a rational function approximation of the corresponding plane wave representation, and it is demonstrated how these boundary conditions can be formulated in terms of fictitious surface densities, governed by second‐order wave equations on the radiating surface. Several well‐established radiation boundary conditions appear as special cases, corresponding to different choices of the coefficients in the rational approximation. The relation between these choices is established, and an explicit formulation in terms of selected directions with ideal transmission is presented. A mechanical interpretation of the fictitious surface densities enables identification of suitable conditions at corners and boundaries of the radiating surface. Numerical examples illustrate excellent results with one or two fictitious layers with suitable corner and boundary conditions. Copyright © 2001 John Wiley & Sons, Ltd.

Surface photovoltage in semiconductors under sub-band-gap illumination: continuous distribution of surface states

Surface photovoltage spectra in semiconductors are analyzed when the sub-band-gap illumination induces the electron transitions from surface states to the conduction band under the assumption that distribution of surface states is continuous. From analysis performed it follows that the fictitious densities of surface states can be induced due to the wavelength dependence of the photoionization capture cross-section of surface states for electrons and by the electron recombination capture cross-section of surface states which depends on the energy position of surface states in the energy gap. The high illumination intensity (laser illumination), which makes completely empty the surface states, can eliminate the fictious surface states when the density of surface states is not very large, the temperature of measurements is low, and the surface potential barrier is high.

Regularized direct and indirect symmetric variational BIE formulations for three-dimensional elasticity

This paper deals with symmetric variational regularized BIE formulations for the mixed elastostatic boundary-value problem. A direct version (i.e. in terms of unknown boundary displacement and tractions) is first established. The formulation expresses the stationarity of an augmented potential energy functional, thus being truly a variational BIE formulation. Then an indirect version (in terms of unknown fictitious densities) is established from the direct one. Both are expressed using at most weakly singular integrals followed by regular integrals, by means of a combined use of indirect regularization and Stokes theorem. The displacements and tractions are required to be C 0, α continuous and piecewise continuous respectively, thus conventional BEM interpolation of any degree may be used. These formulations provide a basis for the numerical solution of 3D elastic problems. The numerical evaluation of the singular integrals that arise in the process is discussed. The formulations presented here provide some understanding of the underlying principles as well as a sound working base for Galerkin boundary element analysis of elastic problems.

Application of modified indirect boundary element method to electromagnetic field problems

This paper proposes a modified indirect boundary element technique and its application to electromagnetic field problems. In the modified indirect boundary element formulation, which is developed following a weighted residual approach, the fictitious source density, which appears in the integral equations of the method, is distributed on a surface which is exteriorly separated from the physical field boundary of the problem. This method does not require the evaluation of singular integrals and produces undeteriorated solutions at geometric discontinuities. By this approach, calculations can be performed using far fewer nodes by the direct boundary element method, and the results obtained have fairly good accuracy.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Electromagnetic scattering by indented screens

The problem of three dimensional electromagnetic scattering from a perfectly conducting screen with a bounded indentation is formulated as a system of boundary integral equations for the electric current density on the cavity wall and the interface between the cavity and free space. It is shown how the fictitious current density on the interface may be eliminated resulting in an integral equation of the second kind for the current density on the cavity wall only, with no integration over the infinite screen. In addition, integral representations are derived that represent the field everywhere in space in terms of the current density on the cavity wall only. Furthermore, asymptotic expressions for the far field are also presented. The equations and representations simplify considerably in the two-dimensional scalar case and results are presented for both TE and TM polarization. >

On the application of a limiting process to the dynamic relaxation analysis of circular membranes, circular plates and spherical shells

Two efficient techniques in the dynamic relaxation (DR) iteration procedure are described: (1) a limiting process technique for the governing equations expressed in polar or spherical coordinates and (2) a new DR-formulation for the time-step proposed by Mikami. The former technique overcomes the mathematical singularity at the coordinate origin without the use of interlacing finite difference meshes. The latter solution technique allows all variables to be evaluated at the same time without the use of fictitious densities. The validity of the use of these two techniques has been demonstrated through the numerical results for the problems of large deflection of axisymmetric circular membranes, large deflection of axisymmetric circular plates, and small deflection of axisymmetric spherical shells.

Boundary element analysis method for 3-D transient eddy current problems based on the second order vector potential formulation

The authors consider a boundary-element method (BEM) for 3-D transient eddy current problems formulated in terms of the second-order vector potential (SOVP). Indirect boundary integral equations are derived using the time-dependent fundamental solution and fictitious source densities. In the numerical implementation, a time-stepping convolution approach is adopted to eliminate the need for volume discretization and domain integration. Numerical results of a benchmark problem demonstrate the validity and accuracy of the technique. The proposed scheme, compared with an alternative volume-based recurring initial condition approach, not only provides considerable reduction in the manpower requirements for modeling, but also permits a more precise solution, especially for 3-D problems.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

A boundary element method for plate bending problems

An indirect boundary element formulation is constructed by using a fundamental solution of the biharmonic operator. It is shown that the fundamental solution and its derivatives can be represented by four functions with two integers. The new representation is not only a simpler representation but also reveals a structure that is exploited in the development of an algorithm based on the analytical integration of line and area integrals. The analytical integral values are used to establish continuity requirements on the unknown fictitious densities. Numerical examples consider the consequences of satisfying and violating the continuity requirements by the unknown fictitious densities. It is also shown that if the fictitious density distributions do not satisfy certain conditions then (he solution can be affected by the choice of the non-dimensionalizing variables. Numerical examples with a variety of boundary conditions demonstrate the effectiveness and limitations of the proposed algorithm.

Modified indirect boundary element technique and its application to electromagnetic potential problems

A modified indirect boundary element formulation is developed following a weighted residual approach. In this formulation, the fictitious source density, which appears in the integral equations of the method, is distributed on a surface which is exteriorly separated from the physical field boundary of the problem. The method does not require the evaluation of singular integrals and produces undeteriorated solutions at geometric discontinuities. The technique is applied to two-dimensional electromagnetic potential problems, and several examples are investigated.