A numerical method for the evaluation of complex integrals

A numerical method for evaluating the complex integralI = frac{1}{2pij} oint frac{B(z)B(z^{-1})}{A(z)A(z^{-1})} frac{dz}{z}along the unit circle in a positive direction (whereAandBare polynomials with real coefficients), is presented in this paper. The method developed in this paper is shown to be obtainable as a FORTRAN program as well as a table form. The results achieved represent significant reduction of the computations compared to other existing methods.     

Improving the numerical evaluation of certain oscillatory integrals

By expanding a component of the non-oscillatory part of the integrand in a series of Laguerre polynomials the numerical approximation of integrals of the type $$\mathop \smallint \limits_0^\infty e^{ - \alpha x^2 } f(x^2 ) \cos (bx) dx$$ is obtained. A method to obtain the optimum value for a scaling parameter is provided.     

On the numerical evaluation of derivatives of Cauchy principal value integrals

Quadrature rules for the approximate evaluation of derivatives of Cauchy principal value integrals (with respect to the free variable inside the integral) can be obtained by formal differentiations of the right sides of the corresponding quadrature rules (without derivatives). The justification of this method, under appropriate conditions, is presented in this short communication. The results of this short communication are also applicable to the numerical evaluation of a class of finite-part integrals and to the numerical solution of singular integrodifferential equations.     

The object-oriented DFT program library S/PHI/nX

In order to simplify the development and implementation process of quantum mechanical algorithms, we developed a set of object-oriented C++ libraries which can exploit modern computer architectures. The libraries are characterized as follows: (i) State-of-the-art computer science techniques have been applied or developed in this work to provide language elements to express algebraic expressions efficiently on modern computer platforms. (ii) Quantum mechanical algorithms are crucial in the field of materials research. The new libraries support the Dirac notation to implement such algorithms in the native language of physicists. (iii) The libraries are completed by elements to express equations of motions efficiently which is required for implementing structural algorithms such as molecular dynamics. Based on these libraries we introduce the DFT program package S/PHI/nX.     

Theoretical study of the Mo–Ru sigma phase

The thermodynamic properties of the Mo–Ru binary σ$\sigma$-phase are investigated using a combination of ab initio   calculations and CALPHAD modeling. Total energy calculations have been performed for the complete set of 32 end-member compounds of a 5-sublattice compound energy model. The internal crystallographic parameters for each end-member compound have been determined by minimising the total energy. A simpler, 3-sublattice model of the Mo–Ru σ$\sigma$-phase is formulated on the basis of calculated total energies. The site occupancy is acquired by minimising the free energy given by the compound energy model. A strong preference of Mo and Ru towards high-coordination sites and icosahedral sites in the Mo–Ru σ$\sigma$-phase is found and analysed in terms of the electronic structure.     

Thermodynamic and ab initio investigation of the Cu–Dy system

A combined ab initio and thermodynamic study of the Cu–Dy system has been performed and a self-consistent thermodynamic database has been obtained. Density functional theory has been applied by using the VASP code in order to obtain the enthalpy of formation at 0 K of intermetallic compounds. Experimental information on the Cu–Dy phase diagram and thermodynamic properties of its alloys have been collected. Using these data, optimized parameters have been obtained by applying the CALPHAD approach. The present results reproduce the experimental data available reasonably well, although some features of the Cu–Dy system need to be further clarified.     

An improved thermodynamic modeling of the Fe-Cr system down to zero kelvin coupled with key experiments

A thermodynamic modeling of the Fe-Cr system down to 0 K is performed on the basis of our recent comprehensive review of this binary system [W. Xiong, M. Selleby, Q. Chen, J. Odqvist, Y. Du, Evaluation of phase equilibria and thermochemical properties in the Fe-Cr system, Crit. Rev. Solid State Mater. Sci. 35 (2010) 125-152]. The model predicts a sign change for the magnetic ordering energy of mixing rather than the enthalpy of mixing in the bcc phase at 0 K. Designed key experiments are performed not only to check the validity of the present modeling but also to assist in understanding the mechanism for spinodal decomposition of the Fe-Cr alloy. Heat capacities and Curie temperatures of several Fe-rich alloys are determined between 320 and 1093 K by employing differential scanning calorimetry. The measured heat capacities are found to be in remarkable agreement with the prediction based on the present modeling. Microstructural patterns and frequency distribution diagrams of Cr are studied in alloys containing 26.65, 31.95, and 37.76 at.% Cr by using atom probe tomography. The observed phase separation results correspond well with our model-predicted boundary for the spinodal decomposition. Interestingly, a horn on the Cr-rich spinodal boundary is predicted below 200 K for the first time. This work demonstrates a way to bridge the ab initio calculations and CALPHAD approach.     

Software tools for high-throughput CALPHAD from first-principles data

We present a set of software tools that largely automate the process of converting ab initio data into thermodynamic databases that can readily be imported into standard thermodynamic modeling softwares. These tools are based on the Special Quasirandom Structures (SQS) formalism, extended to transparently handle, not only traditional fcc, bcc and hcp solid solutions, but also multiple-sublattice structures with possible sublattice disorder. A large database of pre-generated SQS is provided that covers over 30 of the most common multi-sublattice structures and spans the composition ranges of each of their sublattices. In addition, we exploit a theoretically justified and robust method to address the issue of assigning free energies to mechanically unstable “virtual” phases, thus providing a compelling solution to a long-standing problem in CALPHAD modeling, especially in the context of ab initio data. We also propose a simple low-order approximation scheme to include short-range order effects that requires no additional ab initio input. The resulting thermodyamic database seamlessly combines ab initio data (formation energies and, optionally, vibrational free energies) with elemental Scientific Group Thermodata Europe (SGTE) data. The proposed tools provide a clear path to expand the coverage of high-throughput efforts towards non-stoichiometric phases and non-zero temperatures. The generated free energy models can also provide very good starting points to perform complex thermodynamic assessments, especially in cases where the available experimental data poorly constrain some thermodynamic parameters. The Cu-Pt-W phase diagram is calculated as an example.     

Stability of difference Volterra equations: Direct Liapunov method and numerical procedure

A procedure to construct Liapunov functionals for discrete Volterra equations is proposed. Using this procedure stability conditions are derived for general Volterra difference equations. Some applications of the proposed procedure for obtaining stability conditions for linear multistep methods for Volterra integro-differential equations are presented.