Accurate Boundary Integral Method Research Articles

Kernel free boundary integral method for 3D incompressible flow and linear elasticity equations on irregular domains

A second-order accurate kernel-free boundary integral method is presented for Stokes flow and linear elasticity boundary value problems on three-dimensional irregular domains. It solves equations in the framework of boundary integral equations, whose corresponding discrete forms are well-conditioned and solved by the GMRES method. A notable feature of this approach is that the boundary or volume integrals encountered in BIEs are indirectly evaluated by a Cartesian grid-based method, which includes discretizing corresponding simple interface problems with a MAC scheme, correcting discrete linear systems to reduce large local truncation errors near the interface, solving the modified system by a CG method together with an FFT-based Poisson solver. No extra work or special quadratures are required to deal with singular or hyper-singular boundary integrals and the dependence on the analytical expressions of Green’s functions for the integral kernels is completely eliminated. Numerical results are given to demonstrate the efficiency and accuracy of the Cartesian grid-based method.

Boundary integral simulations of boundary layers in linear viscoelastic flow

We investigate the disturbance flow generated by the oscillatory motion of a solid particle in linear viscoelastic (LVE) fluids. We begin with one-mode Maxwell fluids and then add Newtonian components so we can examine a spectrum of Weissenberg numbers and frequency parameters. We compute the fluid dynamics using an accurate boundary integral method with third-order accuracy in space. A unique feature of our method is that we can calculate the stress on the particle surface for a prescribed particle velocity profile. It is well known that a boundary layer develops along an infinite plate under oscillatory motion in a Newtonian fluid. However, when the flow becomes viscoelastic, the boundary layers are fundamentally different from those observed in Newtonian fluids. We perform a series of numerical simulations for the geometry of spheroids, dumbbells, and biconcave disks, and characterize streamlines around these particles and shear stress distributions on the surface of the particles. Specifically, two main results emerge from our investigations: (i) there is a sequence of eddies produced in LVE rather than a single one as in the Newtonian fluid case; and (ii) the eddies develop in the interior of the LVE and barely travel, while in the Newtonian fluid, the eddy is generated on the particle surface and propagates into the fluid. Our numerical findings go beyond the well-known dynamical regimes for Newtonian fluids and highlight the level of complexity of particle dynamics in viscoelastic fluids.

Numerical Study on Viscous Fingering Using Electric Fields in a Hele-Shaw Cell

We investigate the nonlinear dynamics of a moving interface in a Hele-Shaw cell subject to an in-plane applied electric field. We develop a spectrally accurate boundary integral method where a coupled integral equation system is formulated. Although the stiffness due to the high order spatial derivatives can be removed, the long-time simulation is still expensive since the evolving velocity of the interface drops dramatically as the interface expands. We remove this physically imposed stiffness by employing a rescaling scheme, which accelerates the slow dynamics and reduces the computational cost. Our nonlinear results reveal that positive currents restrain finger ramification and promote overall stabilization of patterns. On the other hand, negative currents make the interface more unstable and lead to the formation of thin tail structures connecting the fingers and a small inner region. When no flux is injected, and a negative current is utilized, the interface tends to approach the origin and break up into several drops. We investigate the temporal evolution of the smallest distance between the interface and the origin and find that it obeys an algebraic law $\displaystyle (t_*-t)^b$, where $t_*$ is the estimated pinch-off time.

Regularized hyper-singular boundary integral equation methods for three-dimensional poroelastic problems

This work proposes an accurate hyper-singular boundary integral equation method for dynamic poroelastic problems with Neumann boundary condition in three dimensions and both the direct and indirect methods are adopted to construct combined boundary integral equations. The strongly-singular and hyper-singular integral operators are reformulated into compositions of weakly-singular integral operators and tangential-derivative operators, which allow us to prove the jump relations associated with the poroelastic layer potentials and boundary integral operators in a simple manner. Relying on both the investigated spectral properties of the strongly-singular operators, which indicate that the corresponding eigenvalues accumulate at three points whose values are only dependent on two Lamé constants, and the spectral properties of the Calderón relations of the poroelasticity, we propose low-GMRES-iteration regularized integral equations. Numerical examples are presented to demonstrate the accuracy and efficiency of the proposed methodology by means of a Chebyshev-based rectangular-polar solver.

Design and Fabrication of Silicon-Blazed Gratings for Near-Infrared Scanning Grating Micromirror

Blazed gratings are the critical dispersion elements in spectral analysis instruments, whose performance depends on structural parameters and topography of the grating groove. In this paper, high diffraction efficiency silicon-blazed grating working at 800–2500 nm has been designed and fabricated. By diffraction theory analysis and simulation optimization based on the accurate boundary integral equation method, the blaze angle and grating constant are determined to be 8.8° and 4 μm, respectively. The diffraction efficiency is greater than 33.23% in the spectral range of 800–2500 nm and reach the maximum value of 85.62% at the blaze wavelength of 1180 nm. The effect of platform and fillet on diffraction efficiency is analyzed, and the formation rule and elimination method of the platform are studied. The blazed gratings are fabricated by anisotropic wet etching process using tilted (111) silicon substrate. The platform is minished by controlling etching time and oxidation sharpening process. The fillet radius of the fabricated grating is 50 nm, the blaze angle is 7.4°, and the surface roughness is 0.477 nm. Finally, the blazed grating is integrated in scanning micromirror to form scanning grating micromirror by MEMS fabrication technology, which can realize both optical splitting and scanning. The testing results show that the scanning grating micromirror has high diffraction efficiency in the spectral range of 810–2500 nm for the potential near-infrared spectrometer application.

A boundary integral method to investigate pattern formation in a rotating Hele-Shaw cell with time dependent gap

In this article we present a spectrally accurate boundary integral method to investigate the evolution of a Hele Shaw blob in a rotating, radial cell with time dependent gap. The blob of the viscous fluid is affected by the shearing forces induced by a changing gap distance, however the rotation also has significant influence on the shape of the blob. Using the method of complex moments, we characterize the solution of the moving boundary problem that models the evolution of the blob and investigate the morphology using linear analysis. The results of the linear analysis indicate the de-stabilizing influence of the rotation. In the numerical methods, we develop a spectrally accurate boundary integral technique to explore the complete nonlinear dynamics of the problem. Using the boundary integral simulations we demonstrate that, for a given strength of surface tension, the rotational effect causes a more complex pattern on the fluid interface that involves the formation of narrow fingers and more splitting of the interface. The effect is more pronounced if the angular speed increases. We also prescribe a method for controlling the viscous fingering using insights from linear analysis. Our numerical simulations validate the proposed controlling mechanism.

Nonlinear simulation of vascular tumor growth with chemotaxis and the control of necrosis

In this paper, we develop a sharp interface tumor growth model in two dimensions to study the effect of both the intratumoral structure using a controlled necrotic core and the extratumoral nutrient supply from vasculature on tumor morphology. We first show that our model extends the benchmark results in the literature using linear stability analysis. Then we solve this generalized model numerically using a spectrally accurate boundary integral method in an evolving annular domain, not only with a Robin boundary condition on the outer boundary for the nutrient field which models tumor vasculature, but also with a static boundary condition on the inner boundary for pressure field which models the control of tumor necrosis. The discretized linear systems for both pressure and nutrient fields are shown to be well-conditioned through tracing GMRES iteration numbers. Our nonlinear simulations reveal the stabilizing effects of angiogenesis and the destabilizing ones of chemotaxis and necrosis in the development of tumor morphological instabilities if the necrotic core is fixed in a circular shape. When the necrotic core is controlled in a non-circular shape, the stabilizing effects of proliferation and the destabilizing ones of apoptosis are observed. Finally, the values of the nutrient concentration with its fluxes and the pressure level with its normal derivatives, which are solved accurately at the boundaries, help us to characterize the corresponding tumor morphology and the level of the biophysical quantities on interfaces required in keeping various shapes of the necrotic region of the tumor. Interestingly, we notice that when the necrotic region is fixed in a 3-fold non-circular shape, even if the initial shape of the tumor is circular, the tumor will evolve into a shape corresponding to the 3-fold symmetry of the shape of the fixed necrotic region.

A highly accurate boundary integral method for the elastic obstacle scattering problem

Consider the scattering of a time-harmonic plane wave by a rigid obstacle embedded in a homogeneous and isotropic elastic medium in two dimensions. In this paper, a novel boundary integral formulation is proposed and its highly accurate numerical method is developed for the elastic obstacle scattering problem. More specifically, based on the Helmholtz decomposition, the model problem is reduced to a coupled boundary integral equation with singular kernels. A regularized system is constructed in order to handle the degenerated integral operators. The semi-discrete and full-discrete schemes are studied for the boundary integral system by using the trigonometric collocation method. Convergence is established for the numerical schemes in some appropriate Sobolev spaces. Numerical experiments are presented for both smooth and nonsmooth obstacles to demonstrate the superior performance of the proposed method.

Nonlinear limiting dynamics of a shrinking interface in a Hele-Shaw cell

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Optimal slip velocities of micro-swimmers with arbitrary axisymmetric shapes

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An Accurate HyperSingular Boundary Integral Equation Method for Dynamic Poroelasticity in Two Dimensions

This paper is concerned with the boundary integral equation method for solving the exterior Neumann boundary value problem of dynamic poroelasticity in two dimensions. The main contribution of this work consists of two aspescts: the proposal of a novel regularized boundary integral equation, and the presentation of new regularized formulations of the strongly singular and hyper-singular boundary integral operators. First, turning to the spectral properties of the double-layer operator and the corresponding Calderón relation of the poroelasticity, we propose the novel low-GMRES-iteration integral equation whose eigenvalues are bounded away from zero and infinity. Second, with the help of the classical idea of using Günter derivatives and integration-by-parts, we reformulate the strongly singular and hyper-singular integral operators into combinations of the weakly singular operators and the tangential derivatives. The accuracy and efficiency of the proposed methodology are demonstrated through several numerical examples.

Complex Far-Field Geometries Determine the Stability of Solid Tumor Growth with Chemotaxis

In this paper, we develop a sharp interface tumor growth model to study the effect of the tumor microenvironment using a complex far-field geometry that mimics a heterogeneous distribution of vasculature. Together with different nutrient uptake rates inside and outside the tumor, this introduces variability in spatial diffusion gradients. Linear stability analysis suggests that the uptake rate in the tumor microenvironment, together with chemotaxis, may induce unstable growth, especially when the nutrient gradients are large. We investigate the fully nonlinear dynamics using a spectrally accurate boundary integral method. Our nonlinear simulations reveal that vascular heterogeneity plays an important role in the development of morphological instabilities that range from fingering and chain-like morphologies to compact, plate-like shapes in two dimensions.

Deformation and stability of a viscous electrolyte drop in a uniform electric field

We study the deformation and breakup of an axisymmetric electrolyte drop which is freely suspended in an infinite dielectric medium and subjected to an imposed electric field. The electric potential in the drop phase is assumed small, so that its governing equation is approximated by a linearized Poisson-Boltzmann or modified Helmholtz equation (the Debye-H\"{u}ckel regime). An accurate and efficient boundary integral method is developed to solve the low-Reynolds-number flow problem for the time-dependent drop deformation, in the case of arbitrary Debye layer thickness. Extensive numerical results are presented for the case when the viscosity of the drop and surrounding medium are comparable. Qualitative similarities are found between the evolution of a drop with a thick Debye layer (characterized by the parameter $\chi\ll 1$, which is an inverse dimensionless Debye layer thickness) and a perfect dielectric drop in an insulating medium. In this limit, a highly elongated steady state is obtained for sufficiently large imposed electric field, and the field inside the drop is found to be well approximated using slender body theory. In the opposite limit $\chi\gg 1$, when the Debye layer is thin, the drop behaves as a highly conducting drop, even for moderate permittivity ratio $Q=\epsilon_1/\epsilon_2$, where $\epsilon_1, \epsilon_2$ is the dielectric permittivity of drop interior and exterior, respectively. For parameter values at which steady solutions no longer exist, we find three distinct types of unsteady solution or breakup modes. These are termed conical end formation, end splashing, and open end stretching. The second breakup mode, end splashing, resembles the breakup solution presented in a recent paper [R. B. Karyappa et al., J. Fluid Mech. 754, 550-589 (2014)]. We compute a phase diagram which illustrates the regions in parameter space in which the different breakup modes occur.

Nonlinear simulation of an elastic tumor-host interface

Abstract We develop a computational method for simulating the nonlinear dynamics of an elastic tumor-host interface. This work is motivated by the recent linear stability analysis of a two-phase tumor model with an elastic membrane interface in 2D [47]. Unlike the classic tumor model with surface tension, the elastic interface condition is numerically challenging due to the 4th order derivative from the Helfrich bending energy. Here we are interested in exploring the nonlinear interface dynamics in a sharp interface framework. We consider a curvature dependent bending rigidity (curvature weakening [22]) to investigate metastasis patterns such as chains or fingers that invade the host environment. We solve the nutrient field and the Stokes flow field using a spectrally accurate boundary integral method, and update the interface using a nonstiff semi-implicit approach. Numerical results suggest curvature weakening promotes the development of branching patterns instead of encapsulated morphologies in a long period of time. For non-weakened bending rigidity, we are able to find self-similar shrinking morphologies based on marginally stable value of the apoptosis rate.

A highly accurate boundary integral equation method for surfactant-laden drops in 3D

The presence of surfactants alters the dynamics of viscous drops immersed in an ambient viscous fluid. This is specifically true at small scales, such as in applications of droplet based microfluidics, where the interface dynamics become of increased importance. At such small scales, viscous forces dominate and inertial effects are often negligible. Considering Stokes flow, a numerical method based on a boundary integral formulation is presented for simulating 3D drops covered by an insoluble surfactant. The method is able to simulate drops with different viscosities and close interactions, automatically controlling the time step size and maintaining high accuracy also when substantial drop deformation appears. To achieve this, the drop surfaces as well as the surfactant concentration on each surface are represented by spherical harmonics expansions. A novel reparameterization method is introduced to ensure a high-quality representation of the drops also under deformation, specialized quadrature methods for singular and nearly singular integrals that appear in the formulation are evoked and the adaptive time stepping scheme for the coupled drop and surfactant evolution is designed with a preconditioned implicit treatment of the surfactant diffusion.

A boundary integral method for computing forces on particles in unsteady Stokes and linear viscoelastic fluids

SummaryAccurately characterizing the forces acting on particles in fluids is of fundamental importance for understanding particle dynamics and binding kinetics. Conventional asymptotic solutions may lead to poor accuracy for neighboring particles. In this paper, we develop an accurate boundary integral method to calculate forces exerted on particles for a given velocity field. We focus our study on the fundamental two‐bead oscillating problem in an axisymmetric frame. The idea is to exploit a correspondence principle between the unsteady Stokes and linear viscoelasticity in the Fourier domain such that a unifying boundary integral formulation can be established for the resulting Brinkman equation. In addition to the dimension reduction vested in a boundary integral method, our formulation only requires the evaluation of single‐layer integrals, which can be carried out efficiently and accurately by a hybrid numerical integration scheme based on kernel decompositions. Comparison with known analytic solutions and existing asymptotic solutions confirms the uniform third‐order accuracy in space of our numerical scheme. Copyright © 2016 John Wiley & Sons, Ltd.

Nonlinear simulations of elastic fingering in a Hele-Shaw cell

This work is motivated by the recent experiments of two reacting fluids in a Hele-Shaw cell (Podgorski et al., 2007) and associated linear stability analysis of a curvature weakening model (He et al., 2012). Unlike the classical Hele-Shaw problem posed for moving interfaces with surface tension, the curvature weakening model is concerned with a newly-produced gel-like phase that stiffens the interface, thus the interface is modeled as an elastic membrane with curvature dependent rigidity that reflects geometrically induced breaking of intermolecular bonds. Here we are interested in exploring the long-time interface dynamics in the nonlinear regime. We perform simulations using a spectrally accurate boundary integral method, together with a rescaling scheme to dramatically speed up the intrinsically slow evolution of the interface. We find curvature weakening inhibits tip-splitting and promotes side-branching morphology. At long times, numerical results reveal that there exist nonlinear, stable, self-similarly evolving morphologies.

Dielectric breakdown model for polymer films

This paper describes a modified dielectric breakdown model (mDBM), to predict breakdown pathways in polymer films taking into consideration trapped charge and gaseous voids. Modeling components include: (1) accurate boundary integral equation method (BIEM) based on Green function formulations for grid-less, non-orthogonal, Poisson growth; (2) treatment of kernel singularities; (3) matrix algebra for extensibility, reuse, and rapid iterative computation; and (4) Monte Carlo statistics with inverse probability integral for tree branch growth. 2D and 3D versions of mDBM have been successfully validated against known data and closed-form solutions, viz. charge and capacitance calculations with agreement to within 3% of the theoretical. The versatility of the models has been demonstrated by results of test cases for throughfilm breakdown in the presence of trapped charge, gaseous void, and combination of trapped charge and gaseous void.

A Parallel Adaptive Treecode Algorithm for Evolution of Elastically Stressed Solids

AbstractThe evolution of precipitates in stressed solids is modeled by coupling a quasi-steady diffusion equation and a linear elasticity equation with dynamic boundary conditions. The governing equations are solved numerically using a boundary integral method (BIM). A critical step in applying BIM is to develop fast algorithms to reduce the arithmetic operation count of matrix-vector multiplications. In this paper, we develop a fast adaptive treecode algorithm for the diffusion and elasticity problems in two dimensions (2D). We present a novel source dividing strategy to parallelize the treecode. Numerical results show that the speedup factor is nearly perfect up to a moderate number of processors. This approach of parallelization can be readily implemented in other treecodes using either uniform or non-uniform point distribution. We demonstrate the effectiveness of the treecode by computing the long-time evolution of a complicated microstructure in elastic media, which would be extremely difficult with a direct summation method due to CPU time constraint. The treecode speeds up computations dramatically while fulfilling the stringent precision requirement dictated by the spectrally accurate BIM.

A Fast Accurate Boundary Integral Method for Potentials on Closely Packed Cells

AbstractBoundary integral methods are naturally suited for the computation of harmonic functions on a region having inclusions or cells with different material properties. However, accuracy deteriorates when the cell boundaries are close to each other. We present a boundary integral method in two dimensions which is specially designed to maintain second order accuracy even if boundaries are arbitrarily close. The method uses a regularization of the integral kernel which admits analytically determined corrections to maintain accuracy. For boundaries with many components we use the fast multipole method for efficient summation. We compute electric potentials on a domain with cells whose conductivity differs from that of the surrounding medium. We first solve an integral equation for a source term on the cell interfaces and then find values of the potential near the interfaces via integrals. Finally we use a Poisson solver to extend the potential to a regular grid covering the entire region. A number of examples are presented. We demonstrate that increased refinement is not needed to maintain accuracy as interfaces become very close.