Non-diagonal Mass Matrix Research Articles

Optimal local truncation error method on unfitted Cartesian meshes for solution of 3-D wave and heat equations for heterogeneous materials

In the paper we develop the optimal local truncation error method (OLTEM) with the non-diagonal and diagonal mass matrices on unfitted Cartesian meshes for the 3-D time-dependent wave and heat equations for heterogeneous materials with irregular interfaces. 27-point stencils that are similar to those for linear finite elements are used with OLTEM. There are no unknowns for OLTEM on interfaces between different materials; the structure of the global discrete equations is the same for homogeneous and heterogeneous materials. The calculation of the unknown stencil coefficients is based on the minimization of the local truncation error of the stencil equations, includes the use of the interface conditions and the entire PDE in derivations, and yields the optimal fourth order of accuracy of OLTEM with the non-diagonal mass matrix. This is two order higher than for linear finite elements with similar 27-point stencils. OLTEM with the diagonal mass matrix includes the rigorous calculation of the diagonal mass matrix and provides the optimal second order of accuracy. The 3-D numerical results for heterogeneous materials with irregular interfaces show that at the same number of degrees of freedom, OLTEM is even much more accurate than high-order (up to the 7-th order - the highest order in the commercial code COMSOL) finite elements with much wider stencils. Compared to linear finite elements with similar 27-point stencils, at an accuracy of 0.1% OLTEM decreases the number of degrees of freedom by 550–7300 times. This leads to a huge reduction in computation time.A new 3-D post-processing procedure has been developed with OLTEM for the calculation of the spatial derivatives of time-dependent numerical solutions. The spatial derivatives for each grid point are calculated with the help of one compact 27-point stencil (the same as that in basic computations). In contrast to known post-processing procedures, the new approach includes also the use of the time derivatives of the function, the interface conditions and the original PDE. The spatial derivatives of the OLTEM solutions calculated with the new post-processing procedure are much more accurate compared to those obtained by high-order (up to the 7-th order) finite elements with much wider stencils. At an accuracy of 0.1% for the spatial derivatives, OLTEM decreases the number of degrees of freedom by 1.9⋅106–6⋅109 times compared to linear finite elements. The new post-processing procedure can be equally applied to the calculation of the partial derivatives obtained by other numerical methods as well as to the numerical results for other PDEs.Due to the huge reduction in the computation time compared to existing methods and the use of trivial unfitted Cartesian meshes that are independent of irregular geometry, the proposed technique does not require remeshing for the shape change of irregular geometry and it will be effective for many design and optimization problems as well as for multiscale problems without the scale separation.

Optimal local truncation error method for solution of 2-D elastodynamics problems with irregular interfaces and unfitted Cartesian meshes as well as for post-processing

The optimal local truncation error method (OLTEM) with unfitted Cartesian meshes recently developed for the scalar wave and heat equations for heterogeneous materials is extended to a more complex case of a system of the elastodynamics PDEs. Compact 9-point stencils (similar to those for linear finite elements) are used for OLTEM. Compared to our previous results, a new approach is used for the calculation of the right-hand side of the stencil equations due to body forces. It significantly simplifies the analytical derivations of OLTEM for time-dependent problems. There are no unknowns on interfaces between different materials; the structure of the global semi-discrete equations for OLTEM is the same for homogeneous and heterogeneous materials. For the first time we have also developed OLTEM with the diagonal mass matrix. In contrast to many known approaches with some ad-hoc calculations of the diagonal mass matrix, OLTEM offers a rigorous approach which is a particular case of OLTEM with the non-diagonal mass matrix. Another novelty of the article is a new post-processing procedure for the accurate calculations of stresses. It includes the same compact 9-point stencils as those in basic computations and uses the accelerations and the displacements at the grid points along with the PDEs for the stress calculations. OLTEM yields accurate numerical results for heterogeneous materials with big contrasts in the material properties of different components. Numerical experiments for elastic heterogeneous materials show: a) at the same number of degrees of freedom (dof), OLTEM with unfitted Cartesian meshes is more accurate than linear finite elements with similar stencils and conformed meshes; at the engineering accuracy of 0.1 % for the displacements, OLTEM reduces the number of dof by more than 20 times; at the engineering accuracy of 0.1 % for the stresses, OLTEM with the new post-processing procedure reduces the number of dof by more than 104 times compared to linear finite elements; b) at the same number of dof, OLTEM with unfitted Cartesian meshes is even more computationally efficient than high-order finite elements with much wider stencils and conformed meshes. This will lead to a huge reduction in the computation time for elastodynamics problems solved by OLTEM and will allow the direct computations of some complex wave propagation and structural dynamics problems for heterogeneous materials without the scale separation.

Optimal local truncation error method for 2‐D elastodynamics problems on irregular domains and unfitted Cartesian meshes

AbstractThe optimal local truncation error method (OLTEM) has been developed for 2‐D time‐dependent elasticity problems on irregular domains and trivial unfitted Cartesian meshes. Compact nine‐point uniform and nonuniform stencils (similar to those for linear finite elements on uniform meshes) are used with OLTEM. The stencil coefficients are assumed to be unknown and are calculated by the minimization of the local truncation error. It is shown that the second order of accuracy is the maximum possible accuracy for nine‐point stencils independent of the numerical technique used for their derivations. The special treatment of the Neumann boundary conditions has been developed that does not increase the size of the stencils. The cases of the nondiagonal and diagonal mass matrices are considered for OLTEM. The results of numerical examples are in agreement with the theoretical findings. They also show that due to the minimization of the local truncation error, OLTEM with the nondiagonal mass matrix is much more accurate than linear finite elements and than quadratic and cubic finite elements (up to the engineering accuracy of 0.1%–1%) at the same numbers of degrees of freedom. The proposed numerical technique can be efficiently used for many engineering applications including geomechanics.

A Joint State-Parameter Identification Algorithm of a Structure with Non-Diagonal Mass Matrix Based on UKF with Unknown Mass

Inaccurate mass estimates have been recognized as an important source of uncertainty in structural identification, especially for large-scale structures with old ages. Over the past decades, some identification algorithms for structural states and unknown parameters, including unknown mass, have been proposed by researchers. However, most of these identification algorithms are based on the simplified mechanical model of chain-like structures. For a chain-like structure, the mass matrix and its inverse matrix are diagonal matrices, which simplify the difficulty of identifying the structure with unknown mass. However, a structure with a non-diagonal mass matrix is not of such a simple characteristic. In this paper, an online joint state-parameter identification algorithm based on an Unscented Kalman filter (UKF) is proposed for a structure with a non-diagonal mass matrix under unknown mass using only partial acceleration measurements. The effectiveness of the proposed algorithm is verified by numerical examples of a beam excited by wide-band white noise excitation and a two-story one-span plane frame structure excited by filtered white noise excitation generated according to the Kanai–Tajimi power spectrum. The identification results show that the proposed algorithm can effectively identify the structural state, unknown stiffness, damping and mass parameters of the structures.

Optimally Blended Spectral Elements in Structural Dynamics: Selective Integration and Mesh Distortion

In the field of structural dynamics, spectral finite elements are well known for their appealing approximation properties. Based on a special combination of shape functions and quadrature points, a diagonal mass matrix is obtained. More recently, the so-called optimally blended spectral element method was introduced, which further improves the accuracy but comes at the cost of a non-diagonal mass matrix. In this work, we study and compare the approximation properties of the different spectral and finite element methods. For each method, an [Formula: see text]-version (fine meshes and low-order shape functions) as well as a [Formula: see text]-version (coarse meshes and high-order shape functions) are considered. Special attention is paid to the influence of the quadrature rule used to compute the stiffness matrix and the element distortion on the convergence behavior. The investigations reveal the importance of a correct (full) integration of the stiffness matrix in order to achieve the theoretically predicted convergence rates. However, looking at the full spectrum, novel variants of the method that apply only a single (reduced) quadrature rule for mass and stiffness matrix show a higher accuracy.

Accurate numerical solutions of 2-D elastodynamics problems using compact high-order stencils

A new approach for the increase in the order of accuracy of high-order numerical techniques used for 2-D time-dependent elasticity (structural dynamics and wave propagation) has been suggested on uniform square and rectangular meshes. It is based on the optimization of the coefficients of the corresponding semi-discrete stencil equations with respect to the local truncation error. It is shown that the second order of accuracy of the linear finite elements with 9-point stencils is optimal and cannot be improved. However, we have calculated new 9-point stencils with the second order of accuracy that yield more accurate results than those obtained by the linear finite elements. We have also developed new 25-point stencils (similar to those for the quadratic finite and isogeometric elements) with the optimal sixth order of accuracy for the non-diagonal mass matrix and with the optimal fourth order of accuracy for the diagonal mass matrix. The numerical results are in good agreement with the theoretical findings as well as they show a big increase in accuracy of the new stencils compared with those for the high-order finite elements. At the same number of degrees of freedom, the new approach yields significantly more accurate results than those obtained by the high-order (up to the eighth order) finite elements with the non-diagonal mass matrix. The numerical experiments also show that the new approach with 25-point stencils yields very accurate results for nearly incompressible materials with Poisson’s ratio 0.4999.

Generating functional for the Green’s functions of a two-flavor bosonic model

We examine a simple two-flavor scalar model with a non-diagonal mass matrix. We argue that the conventional definition of the QFT partition function does not allow to evaluate the Green’s functions with respect to the flavor vacuum as it favors only the mass vacuum. By using functional-integral techniques, we derive new generating functional for Green’s functions on the flavor vacuum.

Gravitational waves in doubly coupled bigravity

We consider gravitational waves from the point of view of both their production and their propagation in doubly coupled bigravity in the metric formalism. In bigravity, the two gravitons are coupled by a non-diagonal mass matrix and show birefrigence. In particular, we find that one of the two gravitons propagates with a speed which differs from one. This deviation is tightly constrained by both the gravitational Cerenkov effect and the energy loss of binary pulsars. When emitted from astrophysical sources, the Jordan frame gravitational wave, which is a linear combination of the two propagating gravitons, has a wave form displaying beats. The best prospect of detecting this phenomenon would come from nano-Hertz interferometric experiments.

The fate of U(1)’s at strong coupling in F-theory

U(1) gauge symmetries in F-theory are expected to manifest themselves as codimension three singularities of Calabi-Yau fourfolds. However, some of these are known to become massive at strong coupling via the St\"uckelberg mechanism. In this note, we propose a geometric picture for detecting all U(1)'s, and determining which ones are massive and which ones are massless. We find that massive gauge symmetries show up as codimension three singularities that only admit small, non-K\"ahler, resolutions. Our proposal passes several highly non-trivial tests, including a case with a non-diagonal mass matrix.

Efficient application of Gauss’ principle to generic mechanical systems

This article discusses one of the solutions proposed in the literature to Gauss’ principle known as principle of ‘least constraint’, proposing a clarifying interpretation that allows significant computational improvements when considering its application to problems with non-diagonal mass matrix. The case of non-symmetric mass matrix is briefly discussed as well.

Triangular Spectral Element simulation of two-dimensional elastic wave propagation using unstructured triangular grids

SUMMARY A Triangular Spectral Element Method (TSEM) is presented to simulate elastic wave propagation using an unstructured triangulation of the physical domain. TSEM makes use of a variational formulation of elastodynamics based on unstructured straight-sided triangles that allow enhanced flexibility when dealing with complex geometries and velocity structures. The displacement field is expanded into a high-order polynomial spectral approximation over each triangular subdomain. Continuity between the subdomains of the triangulation is enforced using a multidimensional Lagrangian interpolation built on a set of Fekete points of the triangle. High-order accuracy is achieved by resorting to an analytical computation of the associated internal product and bilinear forms leading to a non-diagonal mass matrix formulation. Therefore, the time stepping involves the solution of a sparse linear algebraic system even in the explicit case. In this paper the accuracy and the geometrical flexibility of the TSEM is explored. Comparison with classical spectral elements on quadrangular grids shows similar results in terms of accuracy and stability even for long simulations. Surface and interface waves are shown to be accurately modelled even in the case of complex topography with the TSEM. Numerical results are presented for 2-D canonical examples as well as more specific problems, such as 2-D elastic wave scattering by a cylinder embedded in an homogeneous half-space. They all illustrate the enhanced geometrical flexibility of the TSEM. This clearly suggests the need of further investigations in computational seismology specifically targeted towards efficient implementations of the TSEM both in the time and the frequency domain.

Superaccurate finite element eigenvalues via a Rayleigh quotient correction

The consistent finite element formulation of the vibration problem generates upper bounds on the corresponding exact eigenvalues but requires the solution of the highly expensive general algebraic eigenproblem Kx = λ Mx with a global matrix M that is of the same sparsity pattern as the global stiffness K. The lumped, diagonal, mass matrix finite element formulation is no longer variationally correct but results in a simplified algebraic eigenproblem of comparable accuracy. We may write the mass matrix as a linear matrix function, M ( γ ) = M 1 + γ M 2 , of parameter γ such that M ( γ = 1 ) is the (diagonal) lumped mass matrix and M ( γ = 0 ) is the consistent mass matrix. It has been shown that an optimal γ exists between these two states which results in superaccurate eigenvalues. What detracts from the appeal of this approach is that the superior accuracy thus achieved comes at the hefty price of having to solve the still general algebraic eigenproblem with a nondiagonal mass matrix. In this note we show that the same superior accuracy can be had by first computing an eigenvector u from Ku = λ Du , in which D = M 1 + M 2 is the lumped, diagonal, mass matrix, and then obtaining the corresponding, superaccurate, eigenvalue from the Rayleigh quotient R [ u ] = u T Ku / u T M ( γ ) u , M ( γ ) = M 1 + γ M 2 for an optimal γ .

Mass-gap equations within the QED framework: I

The author introduces a substructure into the fermion lines of QED Feynman diagrams and find a substantial modification of the running coupling alpha (q2) at energies larger than the substructure mass scale ( Lambda ). The Feldman-Landau ghost vanishes if the number of light charged lepton states has an upper bound. Estimates of this bound are given; an ultraviolet fixed point appears. He applies these results to the radiative generation of charged lepton masses. This mechanism for the generation of fermion masses is analogous to the Schwinger mechanism in the boson sector or to the energy-gap generation in a superconductor. The masses arise non-perturbatively from the self-interaction of radial excitations of a massless bound state with the photon. Correspondingly, the bare masses are assumed to be zero. The coefficients of the resulting mass-gap equations depend on the magnitude of the ultraviolet fixed point. Only the case of asymptotic freedom can give rise to the right behaviour of the coefficients. The masses, obtained numerically, are indeed of the observed order of magnitude. In addition, the mass ratios of the first few states become largely similar to the observed values. That asymptotic freedom behaviour may be due to some kind of grand unification at energies Lambda approximately=1015 GeV; the properties of the heavy hypercolour states are of substantial significance for the whole light mass spectrum. The nondiagonal mass matrix elements are estimated and the effect of diagonalization of the mass matrix is analysed. It does not account for the large electron-muon mass ratio. He discovers that the diagonalization can lead to negative mass roots which sets an upper and physically relevant bound on alpha u, the value of alpha (q2) at q2 approximately= Lambda 2. The paper constitutes a nice example for the case where the desert hypothesis within the standard model does not hold.

An investigation of parallel numerical integration methods for nonlinear dynamics

The criteria for evaluating the robustness of parallel numerical integration algorithms are discussed and several integration algorithms are examined. The results show that the central difference method has high computational efficiency and parallel effectiveness in the case of a diagonal mass and damping matrix and is recommended in such a case. A comparison is also made between the Cholesky decomposition method and a mixed Jacobi/Gauss-Seidel method when they are incorporated in the central difference method for the case of a nondiagonal mass matrix. The results indicate that iterative methods are attractive in parallel computation.

Fluid Mass Matrices for Thin Shell-of-Revolution Tanks

This paper presents the equations for thin shell-of-revolution tanks containing an arbitrary level of fluid. The mass matrix can be used with any finite element or finite difference digital computer program that computes either the dynamic response or the natural frequencies and modes of a thin shell of revolution and that has provision for a nondiagonal mass matrix. The method does not require a numerical finite difference or finite element discretization of the fluid, but instead provides, in series form, a mass matrix that exactly satisfies the governing field equations for the fluid and satisfies the fluid-shell interface condition with a least-squares fit.

Horizontal-quantum-flavor-dynamics approach to the fermion mass spectrum

We show that the important features associated with the fermion mass spectrum can be explained within the framework of an ${[\mathrm{S}\mathrm{U}(2)\phantom{\rule{0ex}{0ex}}\ifmmode\times\else\texttimes\fi{}\mathrm{U}(1)]}_{\mathrm{WS}}$ \ifmmode\times\else\texttimes\fi{} U'(1) gauge theory subject to a generalized generation structure, without assuming any hierarchy among the various Yukawa coupling constants. At the two-generation level, we show that (i) the only allowed nondiagonal mass matrix is of the canonical type which characterizes certain discrete-symmetry models, and (ii) $(\frac{{m}_{u}}{{m}_{c}})\ensuremath{\simeq}{(\frac{{m}_{d}}{{m}_{s}})}^{2}$. However, we argue that the approximate equality (within a factor of 1.5-2) in masses of the two lightest quarks cannot be explained without invoking at least one additional generation of particles. Moreover, the naturalness of ${m}_{u}\ensuremath{\simeq}{m}_{d}$ completely determines the structure of the two charged-quark mass matrices in the six-flavor scheme. It follows then that ${m}_{c}\ensuremath{\gg}{m}_{s}\ensuremath{\gtrsim}{({m}_{c}{m}_{u,d})}^{\frac{1}{2}}$, and the mass scale of the third-generation quarks is fixed by the masses of the two lighter generations to be of order $\frac{{{m}_{c}}^{2}}{{m}_{s}}$. A new relation for the Cabibbo angle, namely ${\ensuremath{\theta}}_{C}\ensuremath{\simeq}{(\frac{{m}_{d}}{{m}_{s}^{\mathrm{eff}}})}^{\frac{1}{2}}$ with ${m}_{s}^{\mathrm{eff}}=\frac{{{m}_{c}}^{3}}{{{m}_{b}}^{2}}$, emerges. We also calculate the main $\mathrm{CP}$-violating parameter associated with the ${K}^{0}\ensuremath{-}{\overline{K}}^{0}$ system to be ${\ensuremath{\epsilon}}_{K}\ensuremath{\simeq}{(\frac{{m}_{s}}{{m}_{c}})}^{3}\ensuremath{\sim}{10}^{\ensuremath{-}3}$ as observed. Finally we discuss the effects of the horizontal gauge symmetry on the leptonic sector and establish the possibility of having a full quarklepton correspondence without conflicting with any known experimental considerations.

Reduction of stiffness and mass matrices

Just as it is often necessary to reduce the size of the stiff­ness matrix in statical structural analysis, the simulta­neous reduction of the nondiagonal mass matrix for natural mode analysis may also be required. The basis for one such reduction technique may follow the procedure used in Ref. 1 for the stiffness matrix, namely, the elimination of coordinates at which no forces are applied.