Integral Formulation Research Articles

Numerical simulation of an extensible capsule using regularized Stokes kernels and overset finite differences

In this paper, we present a novel numerical scheme for simulating deformable and extensible capsules suspended in a Stokesian fluid. The main feature of our scheme is a partition-of-unity (POU) based representation of the surface that enables asymptotically faster computations compared to spherical-harmonics based representations. We use a boundary integral equation formulation to represent and discretize hydrodynamic interactions. The boundary integrals are weakly singular. We use the quadrature scheme based on the regularized Stokes kernels by Tlupova and Beale 2019 (given in [34]). We also use partition-of unity based finite differences that are required for the computation of interfacial forces. Given an N-point surface discretization, our numerical scheme has fourth-order accuracy and O(N) asymptotic complexity, which is an improvement over the O(N2log⁡N) complexity of a spherical harmonics based spectral scheme that uses product-rule quadratures by Veerapaneni et al. 2011 [36]. We use GPU acceleration and demonstrate the ability of our code to simulate the complex shapes with high resolution. We study capsules that resist shear and tension and their dynamics in shear and Poiseuille flows. We demonstrate the convergence of the scheme and compare with the state of the art.

Further Fractional Hadamard Integral Inequalities Utilizing Extended Convex Functions

This work investigates novel fractional Hadamard integral inequalities by utilizing extended convex functions and generalized Riemann-Liouville operators. By carefully using extended integral formulations, we not only find novel inequalities but also improve the accuracy of error bounds related to fractional Hadamard integrals. Our study broadens the applicability of these inequalities and shows that they are useful for a variety of convexity cases. Our results contribute to the advancement of mathematical analysis and provide useful information for theoretical comprehension as well as practical applications across several scientific directions.

Addressing arbitrary body forces in 2D elasticity coupling the Radial Basis Integration Method with boundary elements

A boundary-domain integral formulation inevitably arises when the boundary element method (BEM) is applied for solving the differential equation that governs linear elastostatic problems with body forces. Although the domain integrals introduced by the body forces can be evaluated using internal cells this destroys the boundary-only meshing feature of BEM and makes the integration processes inefficient. This paper shows that these problems can be solved more efficiently without using internal cells by the Radial Basis Integration Method (RBIM) which employs a meshless quadrature obtained by performing boundary-only offline computations. Using RBIM, weakly singular domain integrals can be computed via a simple selective quadrature procedure, whereas, strong singular domain integrals may be computed using two schemes: The first, is based on the same selective quadrature procedure, and the second is based on the singularity separation scheme. The present method, unlike classical radial integration method (RIM), does not have problems in calculating integrals in concave or multiple-connected domains. The results obtained in some 2D linear elastostatic problems with arbitrary body forces show that this method can be as accurate as RIM but less time-consuming than the latter. This method could be applied to other engineering problems involving source terms.

All about the neutron from lattice QCD

I describe how simulations of lattice QCD using the path integral formulation provide the two basic quantum mechanical properties of QCD, its ground state in which correlation functions are calculated, and Fock state wavefunctions between which matrix elements of operators are calculated. Both constructs are stochastic, so unfortunately one gets no intuitive picture or even qualitative understanding of what they look like, nevertheless they contain and display all the subtleties of the quantum field theory. Today, these simulations provide many quantities that are impacting phenomenology and experiments. I illustrate the methods and the steps in the analysis using, as examples, three observables: the isovector charges of the nucleon, the contribution of the quark’s intrinsic spin to the nucleon spin, and the pion-nucleon sigma term.

Two-dimensional nonlocal Eshelby’s inclusion theory: eigenstress-driven formulation and applications

The classical Eshelby’s theory, developed based on local linear elasticity, cannot be applied to inclusion problems that involve nonlocal (long range) elastic effects often observed in micromechanical systems. In this study, we introduce the extension of Eshelby’s inclusion theory to nonlocal elasticity. Starting from Eringen’s integral formulation of nonlocal elasticity, an eigenstress-driven nonlocal Eshelby’s inclusion theory is presented. The eigenstress-driven approach is shown to be a valid mathematical extension of the classical eigenstrain-driven approach in the context of nonlocal inclusion problems. Two individual numerical approaches are developed and applied to simulate inclusion problems and numerically extract the corresponding nonlocal Eshelby tensor. The numerical results obtained from both approaches confirm the validity of the derived nonlocal Eshelby tensor and its ability to capture the non-uniform eigenstress distribution within an elliptic inclusion. These results also help reveal the fundamental difference between the mechanical behaviour of the classical local and the nonlocal inclusion problems. The eigenstress-driven nonlocal inclusion theory could provide the necessary theoretical foundation for the development of homogenization methods of nonlocal heterogeneous media.

Simple, Method of Moments Solution for the Integral Equations for Multiple Dielectric Bodies of Arbitrary Shape in Time Domain

In this work, we present a straightforward and simple method of moments (MOM) solution procedure, with minimum mathematical manipulations, to solve the coupled integral equations for multiple, homogeneous and inhomogeneous, dielectric bodies of arbitrary shape directly in the time domain. The standard surface and volume integral equation formulations are used for homogeneous and inhomogeneous bodies, respectively. The numerical solution procedure does not involve a time-marching process as is usually adopted for time domain problems and seems to be one of the primary reasons for the late-time instabilities as a result of error accumulation. The present solution method is stable for a very long time as evidenced by several representative numerical examples presented for validation.

Characterization of the acoustic pressure waveforms in Rijke tubes with spatially varying heat sources and temperature distributions

In this work, we implement an asymptotic expansion technique that leverages a small perturbation parameter that arises in the context of one-dimensional tubes with open-open end point configurations and spatially varying heat sources. This approach, when paired with a spectral collocation eigensolver, enables us to produce accurate predictions of the pressure mode shapes and frequencies over a wide range of parameters. These include the temperature gain across the heat source, the heat source length and location, and the overall thermal profile. The latter is intended to reproduce different flow heating configurations that emulate Rijke tube characteristics. Specifically, this investigation begins by considering three piecewise representations of the heat source by juxtaposing constant–constant temperatures before and after a heating element whose temperature is prescribed locally using three analytical functions: linear, exponential, and power-law profiles. This is followed by a logistic distribution that can be globally applied to provide a uniformly valid, continuous, and differentiable thermal profile spanning the entire tube, including the heat source element. Our fundamental formulation relies on Green's functions and an integral formulation that enables us to extract all acoustic frequencies analytically. These are found to increase monotonically with successive elevations in the temperature gain across the heat source, retractions of the heat source, length reductions in the heat source, and smoothing of the temperature gain. Along similar lines, the pressure mode shapes are found to exhibit blunter and often linear variations for higher temperature gains, longer heat sources, and upstream displacements of the heat source toward the inlet.

An efficient second-order cell-centered Lagrangian discontinuous Galerkin method for two-dimensional elastic-plastic flows

An efficient second-order cell-centered Lagrangian discontinuous Galerkin (DG) method for solving two-dimensional (2D) elastic-plastic flows with the hypo-elastic constitutive model and von Mises yield condition is presented. First, starting from the governing equations of conserved quantities in the Euler framework, the integral weak formulation of them in the Lagrangian framework is derived. Next, the DG method is used for spatial discretization of both the weak formulation of conserved quantities and the evolution equation of deviatoric stress tensor. The Taylor basis functions defined in the reference coordinates provide the piecewise polynomial expansion of the variables, including the conserved quantities and the deviatoric stress tensor. The vertex velocities and Cauchy stress tensor on the edges are computed using a nodal solver equipped with a variant of Li's new Harten-Lax-van Leer-contact approximate Riemann solver [Li et al., “An HLLC-type approximate Riemann solver for two-dimensional elastic-perfectly plastic model,” J. Comput. Phys. 448, 110675 (2022)], in which the longitudinal wave velocity in the plastic state is modified. Then the vertex velocities and Cauchy stress tensor on the edges are used to compute numerical fluxes. A second-order total variation diminishing Runge–Kutta scheme is used for time discretization of both the governing equations of conserved quantities and the evolution equation of deviatoric stress tensor. After solving the evolution equation of deviatoric stress tensor, a radial return algorithm is performed at the Gauss points of each element according to the von Mises yield condition. And then the coefficients of the DG expansion for the deviatoric stress tensor on each element are modified by a least squares procedure using the deviatoric stress tensors at these Gauss points. To achieve second-order accuracy, the least squares procedure is used for piecewise linear reconstruction of conserved quantities and the deviatoric stress tensor, and the Barth–Jespersen limiter is used to suppress the nonphysical numerical oscillation near the discontinuities. After that, the coefficients of the DG expansion are modified through L2 projection using the reconstructed polynomials. Finally, a second-order cell-centered Lagrangian DG scheme is established. Several tests demonstrate that the new scheme achieves second-order accuracy with good robustness, and that the DG method of updating the deviatoric stress tensor has comparable accuracy and much higher efficiency with mesh refinement compared with previous works.

Integral Formulation for Losses Computation in Tripolar Submarine Cables With Shield Wires, Moisture Barriers, and Armor

Integral Formulation for Losses Computation in Tripolar Submarine Cables With Shield Wires, Moisture Barriers, and Armor

A Time-Stepping Volume Integral Formulation for Nonlinear Field-Circuit Coupled Problems

A Time-Stepping Volume Integral Formulation for Nonlinear Field-Circuit Coupled Problems

Modeling and experiments on the vibro-acoustic analysis of ring stiffened cylindrical shells with internal bulkheads: A comparative study

The vibro-acoustic response of ring stiffened cylindrical shells with internal bulkheads under forced excitation is presented. The numerical analysis model is established using the Jacobi Ritz-Boundary element method. The first-order shear deformation theory, multi-segment technique and artificial spring technology are applied to establish the theoretical model, and the Jacobi orthogonal polynomials are introduced to represent the displacement functions. The Newmark-β integration method is used to obtain the vibration response of the structure, and the time-domain Kirchhoff boundary integral formulation is applied to describe the exterior acoustic field. The vibration and sound radiation of the ring stiffened cylindrical shell under impact and narrow band random loading were measured. The comparative study reveals that the results obtained from the proposed method agree well with the experimental results. The stiffeners and bulkheads have a significant effect on the vibration characteristics of the structure. Additionally, some physical insights into the resonant peaks and sound pressure directivity of the cylindrical shells are provided.

The boundary element method for acoustic transmission with nonconforming grids

Acoustic wave propagation through a homogeneous material embedded in an unbounded medium can be formulated as a boundary integral equation and accurately solved with the boundary element method. The computational efficiency deteriorates at high frequencies due to the increase in mesh size with a fixed number of elements per wavelength and also at high material contrasts due to the ill-conditioning of the linear system. This study presents the design of boundary element methods feasible for nonconforming surface meshes at the material interface. The nonconforming algorithm allows for independent grid generation, improves flexibility, and reduces the degrees of freedom. It works for different boundary integral formulations for Helmholtz transmission problems, operator preconditioning, and coupling with finite element solvers. The extensive numerical benchmarks at canonical configurations and an acoustic foam model confirm the significant improvements in computational efficiency when employing the nonconforming grid coupling in the boundary element method.

Practical Aspects for Applying Picard Iterations to the SIR Model Using Actual Data

The updated version of the Picard method for solving systems of differential equations is employed to solve the SIR system. A local performance of the Picard iteration algorithm combined with the Gauss-Seidel approach is applied to the SIR model. The integral form of the SIR model, in addition to the use of Gauss-Seidel philosophy (using the most recent calculated values), achieved more accuracy in the computational work than those obtained using the differential forms. Documented data regarding the spread of corona virus 19 in the Kingdom of Saudi Arabia region from April to the end of December 2020 were used to calculate the corresponding actual values for the parameters and the initial conditions. Due to efficient management and to obtain representable behaviours, we restricted the size of the study to only 1% of the population. The global characteristics of the integral formulation have affected the calculations accurately. The initial conditions and the model’s parameters are established depending on the documented data. The results illustrate the superiority of the updated Picard formulation over the classical Picard within their domain of convergence. The results of this study illuminate and validate the importance of mathematical modeling. These findings can provide valuable insights into mathematical modeling for those involved in environmental health research, especially those responsible for devising strategic plans.

A novel stabilized nodal integration formulation using particle finite element method for incompressible flow analysis

AbstractIn simulations using the particle finite element method (PFEM) with node‐based strain smoothing technique (NS‐PFEM) to simulate the incompressible flow, spatial and temporal instabilities have been identified as crucial problems. Accordingly, this study presents a stabilized NS‐PFEM‐FIC formulation to simulate an incompressible fluid with free‐surface flow. In the proposed approach, (1) stabilization is achieved by implementing the gradient strain field in place of the constant strain field over the smoothing domains, handling spatial and temporal instabilities in direct nodal integration; (2) the finite increment calculus (FIC) stabilization terms are added using nodal integration, and a three‐step fractional step method is adopted to update pressures and velocities; and (3) a novel slip boundary with the predictor–corrector algorithm is developed to deal with the interaction between the free‐surface flow with rigid walls, avoiding the pressure concentration induced by standard no‐slip condition. The proposed stabilized NS‐PFEM‐FIC is validated via several classical numerical cases (hydrostatic test, water jet impinging, water dam break, and water dam break on a rigid obstacle). Comparisons of all simulations to the experimental results and other numerical solutions reveal good agreement, demonstrating the strong ability of the proposed stabilized NS‐PFEM‐FIC to solve incompressible free‐surface flow with high accuracy and promising application prospects.

Summing over non-singular paths in quantum cosmology

In this paper we provide the DeWitt propagator and its wave function in quantum cosmology using the path integral formulation of quantum gravity. The DeWitt boundary condition is introduced as a way to avoid the Big Bang singularity by positing that the wave function of the Universe vanishes near the Big Bang. However, there is currently no clear definition of the DeWitt boundary condition in the path integral formulation. To address this issue, we use the image method, which eliminates singular paths in the forbidden region of the infinite potential, and apply this method to quantum cosmology based on the Batalin–Fradkin–Vilkovisky formulation of the path integral. We investigate the validity of the image method, and in particular, find that this method is only appropriate when the potential has symmetry with respect to the boundary. We then show that the DeWitt propagator and the DeWitt wave function derived from the image method are consistent with solutions of the Wheeler–DeWitt equation for certain models of quantum cosmology.

Total angular momentum conservation in Ehrenfest dynamics with a truncated basis of adiabatic states.

We show that standard Ehrenfest dynamics does not conserve linear and angular momentum when using a basis of truncated adiabatic states. However, we also show that previously proposed effective Ehrenfest equations of motion [M. Amano and K. Takatsuka, "Quantum fluctuation of electronic wave-packet dynamics coupled with classical nuclear motions," J. Chem. Phys. 122, 084113 (2005) and V. Krishna, "Path integral formulation for quantum nonadiabatic dynamics and the mixed quantum classical limit," J. Chem. Phys. 126, 134107 (2007)] involving the non-Abelian Berry force do maintain momentum conservation. As a numerical example, we investigate the Kramers doublet of the methoxy radical using generalized Hartree-Fock with spin-orbit coupling and confirm that angular momentum is conserved with the proper equations of motion. Our work makes clear some of the limitations of the Born-Oppenheimer approximation when using abinitio electronic structure theory to treat systems with unpaired electronic spin degrees of freedom, and we demonstrate that Ehrenfest dynamics can offer much improved, qualitatively correct results.

Learning domain-independent Green’s function for elliptic partial differential equations

Green’s function characterizes a partial differential equation (PDE) and maps its solution in the entire domain as integrals. Finding the analytical form of Green’s function is a non-trivial exercise, especially for a PDE defined on a complex domain or a PDE with variable coefficients. In this paper, we propose a novel boundary integral network to learn the domain-independent Green’s function, referred to as BIN-G. We evaluate the Green’s function in the BIN-G using a radial basis function (RBF) kernel-based neural network. We train the BIN-G by minimizing the residual of the PDE and the mean squared errors of the solutions to the boundary integral equations for prescribed test functions. By leveraging the symmetry of the Green’s function and controlling refinements of the RBF kernel near the singularity of the Green function, we demonstrate that our numerical scheme enables fast training and accurate evaluation of the Green’s function for PDEs with variable coefficients. The learned Green’s function is independent of the domain geometries, forcing terms, and boundary conditions in the boundary integral formulation. Numerical experiments verify the desired properties of the method and the expected accuracy for the two-dimensional Poisson and Helmholtz equations with variable coefficients.

Solution of a Scalar Two-Dimensional Nonlinear Diffraction Problem for Objects of Arbitrary Shape

In this study, the development, design, and software implementation of the methods for solving the nonlinear diffraction problem were performed. The influence of nonlinear medium defined by the Kerr law on the propagation of a wave passing through an object was examined. The differential and integral formulations of the problem and the nonlinear integral equation were considered. The problem was solved for different bodies with the use of various computational grids. Convergence graphs of the iterative processes were generated. The obtained graphical results were presented. The explicit and implicit methods for solving the integral equation were compared.

Power-duality in path integral formulation of quantum mechanics

Power duality in Feynman's path integral formulation of quantum mechanics is investigated. The power duality transformation consists of a change in coordinate and time variables, an exchange of energy and coupling, and a classical angular momentum replacement. Two physical systems connected by the transformation form a power-dual pair. The propagator (Feynman's kernel) expressed by Feynman's path integral cannot be form-invariant under the transformation, whereas the promotor constructed by modifying Feynman's path integral is found form-invariant insofar as the angular momentum is classical. Upon angular quantization, the power duality breaks down. To save the notion of power duality, the idea of quasi power duality is proposed, which constitutes of an ad hoc angular momentum replacement. The power-dual invariant promotor leads to the quasi-dual invariant Green function. A formula is proposed, which determines the Green function for one of a dual pair by knowing the Green function for the other. As examples, the Coulomb-Hooke dual pair and a family of two-term confinement potentials for a zero-energy state are discussed.

The equivalent source method is an indirect boundary element method with an implicit Voronoi mesh

The Equivalent Source Method (ESM) consists of modeling the acoustical field radiated or scattered from an object by a set of radiating monopole sources located below its surface. While highly efficient for modeling the scattered field by a smooth object, ESM becomes less effective in representing acoustic diffraction from sharp edges. In this article, ESM is understood as an approximation to the single layer integral equation. In this framework, ESM belongs to the family of Indirect Boundary Element Methods (BEM) with an implicit surface mesh slightly below the object surface, and a zero order integration scheme on each element of the mesh. This implicit mesh is well approximated by the Voronoi diagram associated to the source distribution. This formalism establishes a strong link between these two widely used methods (ESM and BEM) and allows the construction of hybrid methods that can combine performances of both methods. On the first hand, a new hybrid method able to model edge diffraction while remaining as computationally efficient as ESM is developed. In this method, labeled “BEM Voronoi”, equivalent sources are positioned on the surface’s object rather than below it and singularities are handled with the integral formulation. On the other hand, a set of new hybrid methods which combines ESM with higher order integration scheme is constructed. Performances of the new methods developed in this paper are numerically compared to each other and to Finite Element simulations (used as a reference) for modeling scattering from an object excited by a plane wave. Each hybrid model is tested on multiple configurations involving two sets of boundary conditions (rigid and elastic objects respectively) successively immersed in two different external environments (water and air) and for both smooth and a sharp-edged objects. The results show that the hybrid method “BEM Voronoi” better handles edge diffraction than conventional ESM, at a much lower computational cost (implementation of BEM Voronoi is 50 times faster than conventional BEM implementation).