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The Kustaanheimo theory of spinor regularization is given a new formulation in terms of geometric algebra. The Kustaanheimo-Stiefel matrix and its subsidiary condition are put in a spinor form directly related to the geometry of the orbit in physical space. A physically significant alternative to the KS subsidiary condition is discussed. Derivations are carried out without using coordinates.
A relatively simple high frequency analysis of electromagnetic scattering from a class of open-ended waveguide discontinuites was developed. The waveguides are composed of perfectly-conducting sections in which the electromagnetic field can be written as the sum of waveguide modes. Junctions are formed at the open end and also within interior regions where different sections are joined. The reflection and transmission properties of each junction are described in terms of a scattering matrix which is determined by combining the modal ray picture with high frequency techniques such as the Geometrical Theory of Diffraction (GTD), the Equivalent Current Method (ECM), and modifications of the Physical Theory of Diffraction (PTD). A new set of equivalent circuits are employed in this ECM analysis which leads to a simple treatment of many types of junction discontinuities. Also, a new procedure is presented to improve the efficiency of the aperture integration at the open end which is required in the PTD procedure for finding the fields radiated from (or coupled to) the open end. Once the scattering matrices are determined, they are then combined using a self-consistent multiple scattering method to obtain the total scattered fields.
This paper states a conjecture which relates Lax-Richmyer stability to the algebraic test of the stability theory of Gustafsson, Kreiss, and Sundstrom (1972), developed for difference approximations to initial boundary value problems where the matrix size J increases linearly with n as n goes to infinity. This corresponds to mesh refinement in both space and time for t = n x delta t = constant.
Although considerable effort has been put into the study of steady state vibration control, there are few methods applicable to transient vibration control of rotorbearing systems. In this paper optimal control theory has been adopted to minimize rotor vibration due to sudden imbalance, e.g., blade loss. The system gain matrix is obtained by choosing the weighting matrices and solving the Riccati equation. Control forces are applied to the system via a feedback loop. A seven mass rotor system is simulated for illustration. A relationship between the number of sensors and the number of modes used in the optimal control model is investigated. Comparisons of responses are made for various configurations of modes, sensors, and actuators. Furthermore, spillover effect is examined by comparing results from collocated and noncollocated sensor configurations. Results show that shaft vibration is significantly attenuated in the closed loop system.
Monte Carlo Simulation and Stochastic FEA are used to predict randomness in the free vibration response of thin unsymmetrically laminated beams. For the present study, it is assumed that randomness in the response is only caused by uncertainties in the ply orientations. The ply orientations may become random or uncertain during the manufacturing process. A new 16-dof beam element, based on the first-order shear deformation beam theory, is used to study the stochastic nature of the natural frequencies. Using variational principles, the element stiffness matrix and mass matrix are obtained through analytical integration. Using a random sequence a large data set is generated, containing possible random ply-orientations. This data is assumed to be symmetric. The stochastic-based finite element model for free vibrations predicts the relation between the randomness in fundamental natural frequencies and the randomness in ply-orientation. The sensitivity derivatives are calculated numerically through an exact formulation. The squared fundamental natural frequencies are expressed in terms of deterministic and probabilistic quantities, allowing to determine how sensitive they are to variations in ply angles. The predicted mean-valued fundamental natural frequency squared and the variance of the present model are in good agreement with Monte Carlo Simulation. Results, also, show that variations between plus or minus 5 degrees in ply-angles can affect free vibration response of unsymmetrically and symmetrically laminated beams.
The deformation properties of the low-lying states in 74 Ge have been investigated using multistep Coulomb excitation. The measurements were carried out with the advanced 𝛾-ray tracking array, GRETINA, and the CHICO2 particle detector. A comprehensive set of 𝐸2 transition and diagonal matrix elements was deduced following an analysis with the semiclassical coupled-channels code GOSIA. The data were compared with results of calculations carried out within the framework of the generalized triaxial rotor model as well as with the configuration interaction shell model and the symmetric rotor model. Results from calculations with covariant density functional theory were used to construct a five-dimensional collective Hamiltonian for further comparisons with the data. Collectively, the calculations provide an accurate reproduction of the experimental matrix elements and further support an understanding in terms of the coexistence of two axially asymmetric shapes. In conclusion, this leads to an overall interpretation of the underlying structure of 74 Ge requiring triaxiality, as is also the case in the neighboring even-mass Ge isotopes.
The time-dependent exchange–correlation potential has the unusual task of directing fictitious non-interacting electrons to move with exactly the same probability density as true interacting electrons. This has intriguing implications for its structure, especially in the non-perturbative regime, leading to step and peak features that cannot be captured by bootstrapping any ground-state functional approximation. Here, we review what has been learned about these features in the exact exchange–correlation potential of time-dependent density functional theory in the past decade or so and implications for the performance of simulations when electrons are driven far from any ground state.
We use the method of Lightcone Conformal Truncation (LCT) to obtain form factors and spectral densities of local operators $\mathcal{O}$ in $\phi^4$ theory in two dimensions. We show how to use the Hamiltonian eigenstates from LCT to obtain form factors that are matrix elements of a local operator $\mathcal{O}$ between single-particle bra and ket states, and we develop methods that significantly reduce errors resulting from the finite truncation of the Hilbert space. We extrapolate these form factors as a function of momentum to the regime where, by crossing symmetry, they are form factors of $\mathcal{O}$ between the vacuum and a two-particle asymptotic scattering state. We also compute the momentum-space time-ordered two-point functions of local operators in LCT. These converge quickly at momenta away from branch cuts, allowing us to indirectly obtain the time-ordered correlator and the spectral density at the branch cuts. We focus on the case where the local operator $\mathcal{O}$ is the trace Θ of the stress tensor.
This study addresses matrix-dominated failures in carbon fiber/polymer matrix composite laminates in a cross-ply lay-up. The events of interest are interlaminar fracture in the form of transverse cracks in the 90' plies and longitudinal splitting in the 0 deg plies and interlaminar fracture in the form of 0/90 delamination. These events were observed using various nondestructive evaluation (NDE) techniques during static tensile tests. Acoustic emission (AE) x radiography, and edge view microscopy were the principal ones utilized in a real-time environment. A comparison of the NDE results with an analytical model based on the classical linear fracture mechanics concept of strain energy release rate as a criterion for crack growth was performed. The virtual crack closure theory was incorporated with a finite element model to generate strain energy release rate curves for the analytical case. Celion carbon fiber/polyimide matrix (G30-500/PMR-15) was the material tested with cross-ply lay-ups of (0(2)/90(6))s and (0(4)/90(4))s. The test specimens contained thermally induced cracks caused by the high-temperature processing. The analytical model was updated to compensate for the initial damage and to study further accumulation by taking into account the crack interactions. By correlating the experimental and analytical data, the critical energy release rates were found for the observable events of interest.
A major problem in using SVD (singular-value decomposition) as a tool in determining the effective rank of a perturbed matrix is that of distinguishing between significantly small and significantly large singular values to the end, conference regions are derived for the perturbed singular values of matrices with noisy observation data. The analysis is based on the theories of perturbations of singular values and statistical significance test. Threshold bounds for perturbation due to finite-precision and i.i.d. random models are evaluated. In random models, the threshold bounds depend on the dimension of the matrix, the noisy variance, and predefined statistical level of significance. Results applied to the problem of determining the effective order of a linear autoregressive system from the approximate rank of a sample autocorrelation matrix are considered. Various numerical examples illustrating the usefulness of these bounds and comparisons to other previously known approaches are given.
Here, in this article, we propose an algorithm for approximating the action of φ $-$ functions of matrices against vectors, which is a key operation in exponential time integrators. In particular, we consider matrices with Kronecker sum structure, which arise from problems admitting a tensor product representation. The method is based on quadrature approximations of the integral form of the φ $-$ functions combined with a scaling and modified squaring method. Owing to the Kronecker sum representation, only actions of 1D matrix exponentials are needed at each quadrature node and assembly of the full matrix can be avoided. Additionally, we derive a priori bounds for the quadrature error, which show that, as expected by classical theory, the rate of convergence of our method is supergeometric. Guided by our analysis, we construct a fast and robust method for estimating the optimal scaling factor and number of quadrature nodes that minimizes the total cost for a prescribed error tolerance. We investigate the performance of our algorithm by solving several linear and semilinear time-dependent problems in 2D and 3D. The results show that our method is accurate and orders of magnitude faster than the current state-of-the-art.
One-particle Green’s function methods can model molecular and solid spectra at zero or non-zero temperatures. One-particle Green’s functions directly provide electronic energies and one-particle properties, such as dipole moment. However, the evaluation of two-particle properties, such as $\langle$S 2 $\rangle$ and $\langle$N 2 $\rangle$, can be challenging because they require a solution of the computationally expensive Bethe–Salpeter equation to find two-particle Green’s functions. We demonstrate that the solution of the Bethe–Salpeter equation can be completely avoided. Applying the thermodynamic Hellmann–Feynman theorem to self-consistent one-particle Green’s function methods, we derive expressions for two-particle density matrices in a general case and provide explicit expressions for GF2 and GW methods. Such density matrices can be decomposed into an antisymmetrized product of correlated one-electron density matrices and the two-particle electronic cumulant of the density matrix. Cumulant expressions reveal a deviation from ensemble representability for GW, explaining its known deficiencies. We analyze the temperature dependence of $\langle$S 2 $\rangle$ and $\langle$N 2 $\rangle$ for a set of small closed-shell systems. Interestingly, both GF2 and GW show a non-zero spin contamination and a non-zero fluctuation of the number of particles for closed-shell systems at the zero-temperature limit.
We expand Hilbert series technologies in effective field theory for the inclusion of massive particles, enabling, among other things, the enumeration of operator bases for non-linearly realized gauge theories. We find that the Higgs mechanism is manifest at the level of the Hilbert series, as expected for the partition function of an S-matrix that is subject to the Goldstone equivalence theorem. In addition to massive vectors, we detail how other massive, spinning particles can be studied with Hilbert series; in particular, we spell out the ingredients for massive gravity in general spacetime dimensions. Further methodology is introduced to enable Hilbert series to capture the effect of spurion fields acquiring vevs. We apply the techniques to the Higgs Effective Field Theory (HEFT), providing a systematic enumeration of its operator basis. This is achieved both from a direct and a custodial symmetry spurion-based approach; we compare and contrast the two approaches, and our results to those appearing in previous literature.
In this work, we report quantum Monte Carlo calculations of weak transitions in A≤10 nuclei, based on the Norfolk two- and three-nucleon chiral interactions, and associated one- and two-body axial currents. Furthermore, we find that the contribution from two-body currents is at the 2–3% level, with the exception of matrix elements entering the rates of 8Li, 8B, and 8He β decay. These matrix elements are suppressed in impulse approximation based on the (leading order) Gamow Teller transition operator alone; two-body currents provide a 20–30% correction, which is, however, insufficient to bring theory in agreement with experimental data. For the other transitions, the agreement with the data is satisfactory, and the results exhibit a negligible to mild model dependence when different combinations of Norfolk interactions are utilized to construct the nuclear wave functions. We report a complete study of two-body weak transition densities which reveals the expected universal behavior of two-body currents at short distances throughout the range of A=3 to A=10 systems considered here.
Pore-scale modeling has limited applicability at large scales due to its high computational cost. One common approach to upscale pore-scale models is the use of effective medium theories, which homogenize small-scale features in a porous structure and characterize the medium by macroscale properties (e.g., permeability) and equations (e.g., Darcy’s law). However, there are classes of physical processes for which effective medium approximations may become inaccurate, e.g., mineral precipitation and clogging during reactive transport. We have developed a deep learning upscaling framework, in which pore-scale modeling is directly employed in macroscale systems, without relying on effective medium approximations. The upscaling framework is first developed for general multiscale systems and then applied to modeling reactive transport with mineral precipitation in the altered layer in fracture-matrix structures. Solute transport from the fractures to the matrix is modeled as a wall boundary condition for the fractures, which, in turn, is predicted by recurrent neural networks using the concentration histories at the fracture-matrix boundary. Specifically, we consider a meter-scale fracture network embedded in sandstones, where the smallest feature is at the micron scale. Here the proposed framework allows us to span five orders of magnitude in length scales by capturing mineral precipitation in the altered layer of the rock matrix at the pore scale across the entire meter-scale fracture network.
Triply-periodic networks are among the most complex and functionally valuable self-assembled morphologies, yet they form in nearly every class of biological and synthetic soft matter building blocks. In contrast to simpler assembly motifs – spheres, cylinders, layers – networks require molecules to occupy variable local environments, confounding attempts to understand their formation. Here, we examine the double-gyroid network phase by using a geometric formulation of the strong stretching theory of block copolymer melts, a prototypical soft self-assembly system. The theory establishes the direct link between molecular packing, assembly thermodynamics and the medial map, a generic measure of the geometric center of complex shapes. We show that “medial packing” is essential for stability of double-gyroid in strongly-segregated melts, reconciling a long-standing contradiction between infinite- and finite-segregation theories. Additionally, we find a previously unrecognized non-monotonic dependence of network stability on the relative entropic elastic stiffness of matrix-forming to tubular-network forming blocks. The composition window of stable double-gyroid widens for both large and small elastic asymmetry, contradicting intuitive notions that packing frustration is localized to the tubular domains. This study demonstrates the utility of optimized medial tessellations for understanding soft-molecular assembly and packing frustration via an approach that is readily generalizable far beyond gyroids in neat block copolymers.
Abstract We performed a set of numerical simulations to characterize the interplay of fracture network topology, upscaling, and mesh refinement on flow and transport properties in fractured porous media. We generated a set of generic three-dimensional discrete fracture networks at various densities, where the radii of the fractures were sampled from a truncated power-law distribution, and whose parameters were loosely based on field site characterizations. We also considered five network densities, which were defined using a dimensionless version of density based on percolation theory. Once the networks were generated, we upscaled them into a single continuum model using the upscaled discrete fracture matrix model presented by Sweeney et al. (2019). We considered steady, isothermal pressure-driven flow through each domain and then simulated conservative, decaying, and adsorbing tracers using a pulse injection into the domain. For each simulation, we calculated the effective permeability and solute breakthrough curves as quantities of interest to compare between network realizations. We found that selecting a mesh resolution such that the global topology of the upscaled mesh matches the fracture network is essential. If the upscaled mesh has a connected pathway of fracture (higher permeability) cells but the fracture network does not, then the estimates for effective permeability and solute breakthrough will be incorrect. False connections cannot be eliminated entirely, but they can be managed by choosing appropriate mesh resolution and refinement for a given network. Adopting octree meshing to obtain sufficient levels of refinement leads to fewer computational cells (up to a 90% reduction in overall cell count) when compared to using a uniform resolution grid and can result in a more accurate continuum representation of the true fracture network.
Developed over the past decade, TeraChem is an electronic structure and ab initio molecular dynamics software package designed from the ground up to leverage graphics processing units (GPUs) to perform large-scale ground and excited state quantum chemistry calculations in the gas and the condensed phase. TeraChem’s speed stems from the reformulation of conventional electronic structure theories in terms of a set of individually optimized high-performance electronic structure operations (e.g., Coulomb and exchange matrix builds, one- and two-particle density matrix builds) and rank-reduction techniques (e.g., tensor hypercontraction). Recent efforts have encapsulated these core operations and provided language-agnostic interfaces. Finally, this greatly increases the accessibility and flexibility of TeraChem as a platform to develop new electronic structure methods on GPUs and provides clear optimization targets for emerging parallel computing architectures.