| 1. |
- Edvardsson, Sverker, et al.
(author)
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corr3p_tr : A particle approach for the general three-body problem
- 2016
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In: Computer Physics Communications. - : Elsevier BV. - 0010-4655 .- 1879-2944. ; 200, s. 259-273
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Journal article (peer-reviewed)abstract
- This work presents a convenient way to solve the non-relativistic Schrodinger equation numerically for a general three-particle system including full correlation and mass polarization. Both Coulombic and non-Coulombic interactions can be studied. The eigensolver is based on a second order dynamical system treatment (particle method). The Hamiltonian matrix never needs to be realized. The wavefunction evolves towards the steady state solution for which the Schrodinger equation is fulfilled. Subsequent Richardson extrapolations for several meshes are then made symbolically in matlab to obtain the continuum solution. The computer C code is tested under Linux 64 bit and both double and extended precision versions are provided. Test runs are exemplified and, when possible, compared with corresponding values in the literature. The computer code is small and self contained making it unusually simple to compile and run on any system. Both serial and parallel computer runs are straight forward. Program summary Program title: corr3p_tr Catalogue identifier: AEYR_v1_0 Program summary URL: http://cpc.cs.qub.ac.uk/summaries/AEYR_v1_0.html Program obtainable from: CPC Program Library, Queen's University, Belfast, N. Ireland Licensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.ukilicence/licence.html No. of lines in distributed program, including test data, etc.: 15025 No. of bytes in distributed program, including test data, etc.: 156430 Distribution format: tar.gz Programming language: ANSI C. Computer: Linux 64bit PC. Operating system: Linux 64bit. RAM: 300 M bytes Classification: 2.7, 2.8, 2.9. Nature of problem: The Schrodinger equation for an arbitrary three -particle system is solved using finite differences and a fast particle method for the eigenvalue problem [20, 21, 23]. Solution method: A fast eigensolver is applied (see Appendix). This solver works for both symmetrical and nonsymmetrical matrices (which opens up for more accurate nonsymmetrical finite difference expressions to be applied at the boundaries). The three-particle Schrodinger equation is transformed in two major steps. First step is to introduce the function Q(r(1), (r)2, mu) = r(1)r(2)(1 - mu(2))phi(r(1), r(2), mu), where mu = cos (0(12)). The cusps (r(1) = r(2), mu = 1) are then transformed into boundary conditions. The derivatives of Qare then continuous in the whole computational space and thus the finite difference expressions are well defined. Three-particle coalescence (r(1) = r(2) = 0, mu) is treated in the same way. The second step is to replace Q(r(1), r(2), mu) with (2,root x(1)x(2))(-1)Q(x(1) x(2), mu). The space (x(1), x(2), mu) is much more appropriate for a finite difference approach since the square roots x(1) = root r(1), x(2) = root r(2) allow the boundaries to be much further out. The non-linearity of the x-grid also leads to a finer description near the nucleus and a coarser one further out thus resulting in a saving of grid points. Also, in contrast to the usual variable r(12), we have instead used mu which is an independent variable. This simplifies the mathematics and numerical treatments. Several different grids can naturally run completely independent of each other thus making parallel computations trivial. From several grid results the physical property of interest is extrapolated to continuum space. The extrapolations are made in a matlab m-script where all computations can be made symbolically so the loss of decimal figures are minimized during this process. The computer code, including correlation effects and mass polarization, is highly optimized and deals with either triangular or quadratic domains in (x(1), x(2)). Restrictions: The amount of CPU time may become unreasonable for states needing boundary conditions very far beyond the origin. Also if the condition number of the corresponding Hamiltonian matrix is very high, the number of iterations will grow. The use of double precision computations also puts a limit on the accuracy of extrapolated results to about 6-7 decimal figures. Unusual features: The numerical solver is based on a particle method presented in [20, 21, 23]. In the Appendix we provide specific details of dealing with eigenvalue problems. The program uses a 64 bit environment (Linux 64bit). Parallel runs can be made conveniently through a simple bash script. Additional comments: The discretized wavefunction is complete on every given grid. New interactions can therefore conveniently be added to the Hamiltonian without the need to seek for an appropriate basis set. Running time: Given a modern CPU such as Intel core i5 and that the outer boundary conditions of r(1) and r(2) is limited to, say 16 atomic units, the total CPU time of totally 10 grids of a serial run is typically limited to a few minutes. One can then expect about 6-7 correct figures in the extrapolated eigenvalue. A single grid of say h(1) = h(2) = h(3) = 1/16 converges in less than 1 s (with an error in the eigenvalue of about 1 percent). Parallel runs are possible and can further minimize CPU times for more demanding tasks. References: [20] S. Edvardsson, M. Gulliksson, and J. Persson.). Appl. Mech. ASME, 79 (2012) 021012. [21] S. Edvardsson, M. Neuman, P Edstrom, and H. Olin. Comp. Phys. Commun. 197 (2015) 169. [23] M. Neuman, S. Edvardsson, P. Edstrom, Opt. Lett. 40 (2015) 4325.
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| 2. |
- Edvardsson, Sverker, et al.
(author)
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Solving equations through particle dynamics
- 2015
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In: Computer Physics Communications. - : Elsevier. - 0010-4655 .- 1879-2944. ; 197, s. 169-181
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Journal article (peer-reviewed)abstract
- The present work evaluates a recently developed particle method (DFPM). The basic idea behind this method is to utilize a Newtonian system of interacting particles that through dissipation solves mathematical problems. We find that this second order dynamical system results in an algorithm that is among the best methods known. The present work studies large systems of linear equations. Of special interest is the wide eigenvalue spectrum. This case is common as the discretization of the continuous problem becomes dense. The convergence rate of DFPM is shown to be in parity with that of the conjugate gradient method, both analytically and through numerical examples. However, an advantage with DFPM is that it is cheaper per iteration. Another advantage is that it is not restricted to symmetric matrices only, as is the case for the conjugate gradient method. The convergence properties of DFPM are shown to be superior to the closely related approach utilizing only a first order dynamical system, and also to several other iterative methods in numerical linear algebra. The performance properties are understood and optimized by taking advantage of critically damped oscillators in classical mechanics. Just as in the case of the conjugate gradient method, a limitation is that all eigenvalues (spring constants) are required to be of the same sign. DFPM has no other limitation such as matrix structure or a spectral radius as is common among iterative methods. Examples are provided to test the particle algorithm’s merits and also various performance comparisons with existent numerical algorithms are provided.
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| 3. |
- Neuman, Magnus, et al.
(author)
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Solving the radiative transfer equation with a mathematical particle method
- 2015
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In: Optics Letters. - : Optics Info Base, Optical Society of America. - 0146-9592 .- 1539-4794. ; 40:18, s. 4325-4328
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Journal article (peer-reviewed)abstract
- We solve the radiative transfer equation (RTE) using a recently proposed mathematical particle method, originally developed for solving general functional equations. We show that, in the case of the RTE, it gives several advantages, such as handling arbitrary boundary conditions and phase functions and avoiding numerical instability in strongly forward-scattering media. We also solve the RTE, including fluorescence, and an example is shown with a fluorescence cascade where light is absorbed and emitted in several steps. We show that the evaluated particle method is straightforward to implement, which is in contrast with many traditional RTE solvers, but a potential drawback is the tuning of the method parameters.
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| 4. |
- Persson, Johan
(author)
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On dynamic crack growth in discontinuous materials
- 2015
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Doctoral thesis (other academic/artistic)abstract
- In this thesis work numerical procedures are developed for modeling dynamic fracture of discontinuous materials, primarily materials composed of a load-bearing network. The models are based on the Newtonian equations of motion, and does not require neither stiffness matrices nor remeshing as cracks form and grow. They are applied to a variety of cases and some general conclusions are drawn. The work also includes an experimental study of dynamic crack growth in solid foam. The aims are to deepen the understanding of dynamic fracture by answering some relevant questions, e.g. What are the major sources of dissipation of potential energy in dynamic fracture? What are the major differences between the dynamic fracture in discontinuous network materials as compared to continuous materials? Is there any situation when it would be possible to utilize a homogenization scheme to model network materials as continuous? The numerical models are compared with experimental results to validate their ability to capture the relevant behavior, with good results. The only two plausible dissipation mechanisms are energy spent creating new surfaces, and stress waves, where the first dominates the behavior of slow cracks and the later dominates fast cracks. In the numerical experiments highly connected random fiber networks, i.e. structures with short distance between connections, behaves phenomenologically like a continuous material whilst with fewer connections the behavior deviates from it. This leads to the conclusion that random fiber networks with a high connectivity may be treated as a continuum, with appropriately scaled material parameters. Another type of network structures is the ordered networks, such as honeycombs and semi-ordered such as foams which can be viewed as e.g. perturbed honeycomb grids. The numerical results indicate that the fracture behavior is different for regular honeycombs versus perturbed honeycombs, and the behavior of the perturbed honeycomb corresponds well with experimental results of PVC foam.
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