Accurate, high-order representation of complex three-dimensional surfaces via Fourier continuation analysis

We present a new method for construction of high-order parametrizations of surfaces: starting from point clouds, the method we propose can be used to produce full surface parametrizations (by sets of local charts, each one representing a large surface patch – which, typically, contains thousands of the points in the original point-cloud) for complex surfaces of scientific and engineering relevance. The proposed approach accurately renders both smooth and non-smooth portions of a surface: it yields super-algebraically convergent Fourier series approximations to a given surface up to and including all points of geometric singularity, such as corners, edges, conical points, etc. In view of their C^∞ smoothness (except at true geometric singularities) and their properties of high-order approximation, the surfaces produced by this method are suitable for use in conjunction with high-order numerical methods for boundary value problems in domains with complex boundaries, including PDE solvers, integral equation solvers, etc. Our approach is based on a very simple concept: use of Fourier analysis to continue smooth portions of a piecewise smooth function into new functions which, defined on larger domains, are both smooth and periodic. The “continuation functions” arising from a function f converge super-algebraically to f in its domain of definition as discretizations are refined. We demonstrate the capabilities of the proposed approach for a number of surfaces of engineering relevance.     

High-order unconditionally stable FC-AD solvers for general smooth domains I. Basic elements

We introduce a new methodology for the numerical solution of Partial Differential Equations in general spatial domains: our algorithms are based on the use of the well-known Alternating Direction Implicit (ADI) approach in conjunction with a certain “Fourier continuation” (FC) method for the resolution of the Gibbs phenomenon. Unlike previous alternating direction methods of order higher than one, which can only deliver unconditional stability for rectangular domains, the present high-order algorithms possess the desirable property of unconditional stability for general domains; the computational time required by our algorithms to advance a solution by one time-step, in turn, grows in an essentially linear manner with the number of spatial discretization points used. In this paper we demonstrate the FC-AD methodology through a variety of examples concerning the Heat and Laplace Equations in two and three-dimensional domains with smooth boundaries. Applications of the FC-AD methodology to Hyperbolic PDEs together with a theoretical discussion of the method will be put forth in a subsequent contribution. The numerical examples presented in this text demonstrate the unconditional stability and high-order convergence of the proposed algorithms, as well the very significant improvements they can provide (in one of our examples we demonstrate a one thousand improvement factor) over the computing times required by some of the most efficient alternative general-domain solvers.     

A spectral FC solver for the compressible Navier–Stokes equations in general domains I: Explicit time-stepping

We present a Fourier continuation (FC) algorithm for the solution of the fully nonlinear compressible Navier–Stokes equations in general spatial domains. The new scheme is based on the recently introduced accelerated FC method, which enables use of highly accurate Fourier expansions as the main building block of general-domain PDE solvers. Previous FC-based PDE solvers are restricted to linear scalar equations with constant coefficients. The FC methodology presented in this text thus constitutes a significant generalization of the previous FC schemes, as it yields general-domain FC solvers for nonlinear systems of PDEs. While not restricted to periodic boundary conditions and therefore applicable to general boundary value problems on arbitrary domains, the proposed algorithm inherits many of the highly desirable properties arising from rapidly convergent Fourier expansions, including high-order convergence, essentially spectrally accurate dispersion relations, and much milder CFL constraints than those imposed by polynomial-based spectral methods—since, for example, the spectral radius of the FC first derivative grows linearly with the number of spatial discretization points. We demonstrate the accuracy and optimal parallel efficiency of the algorithm in a variety of scientific and engineering contexts relevant to fluid-dynamics and nonlinear acoustics.     

Modeling and computation of two phase geometric biomembranes using surface finite elements

Biomembranes consisting of multiple lipids may involve phase separation phenomena leading to coexisting domains of different lipid compositions. The modeling of such biomembranes involves an elastic or bending energy together with a line energy associated with the phase interfaces. This leads to a free boundary problem for the phase interface on the unknown equilibrium surface which minimizes an energy functional subject to volume and area constraints. In this paper we propose a new computational tool for computing equilibria based on an L² relaxation flow for the total energy in which the line energy is approximated by a surface Ginzburg–Landau phase field functional. The relaxation dynamics couple a nonlinear fourth order geometric evolution equation of Willmore flow type for the membrane with a surface Allen–Cahn equation describing the lateral decomposition. A novel system is derived involving second order elliptic operators where the field variables are the positions of material points of the surface, the mean curvature vector and the surface phase field function. The resulting variational formulation uses H¹ spaces, and we employ triangulated surfaces and H¹ conforming quadratic surface finite elements for approximating solutions. Together with a semi-implicit time discretization of the evolution equations an iterative scheme is obtained essentially requiring linear solvers only. Numerical experiments are presented which exhibit convergence and the power of this new method for two component geometric biomembranes by computing equilibria such as dumbbells, discocytes and starfishes with lateral phase separation.     

Fast elliptic solvers in cylindrical coordinates and the Coulomb collision operator

In this paper, we describe a new class of fast solvers for separable elliptic partial differential equations in cylindrical coordinates (r, θ, z) with free-space radiation conditions. By combining integral equation methods in the radial variable r with Fourier methods in θ and z, we show that high-order accuracy can be achieved in both the governing potential and its derivatives. A weak singularity arises in the Fourier transform with respect to z that is handled with special purpose quadratures. We show how these solvers can be applied to the evaluation of the Coulomb collision operator in kinetic models of ionized gases.     

Micropatterning of perfluoroalkyl self-assembled monolayers for arraying proteins and cells on chips

Organosilane self-assembled monolayers (SAMs) with perfluoroalkyl groups (R_f) on glass surfaces were used for arraying proteins and cells on chips. Quartz crystal microbalance measurements confirmed the inhibition of protein adsorption on R_f-SAM-modified surfaces and showed efficient adsorption on hydroxyl-, carboxyl-, and amino group-modified surfaces. The characteristics of R_f-modified surfaces were evaluated using solvent contact angle measurement and Fourier transform infrared (FTIR) spectroscopy. The R_f surface was highly water- and oil-resistant, as inferred from the contact angles of water, oleic acid, and hexadecane. Specific peaks of IR spectra were detected in the region from 1160 to 1360 cm⁻¹. Etching with dry plasma completely exfoliated the R_f-SAM, exposing the underlying intact glass surface. Modification conditions were optimized using contact angle and FTIR measurements. After dry plasma processing, the contact angles of all solvents became undetectable, and the IR peaks disappeared. Micrometer scale protein and cell patterns can be fabricated using the proposed method. Protein adsorption on micropatterned R_f-SAM-modified chips was evaluated using fluorescence analysis; protein adsorption was easily controlled by patterning R_f-SAM. PC12 and HeLa cells grew well on micropatterned R_f-SAM-modified chips. Micropatterning of R_f-SAM by dry plasma treatment with photolithography is useful for the spatial arrangement of proteins and cells.     

A high-order cut-cell method for numerical simulation of hypersonic boundary-layer instability with surface roughness

Laminar-turbulent transition of hypersonic boundary layers can be affected significantly by the existence of surface roughness. Currently many important mechanisms of roughness-induced transition are not well understood. In recent years, direct numerical simulation (DNS) has been extensively applied for investigating instability and transition mechanisms of hypersonic boundary layers. Most of the past DNS studies, however, have been based on body-fitted grids for smooth surfaces without roughness. Due to complex geometry of embedded roughness, the use of body-fitted grids can be very difficult for flow with arbitrary surface roughness. In this paper, we present a new high-order cut-cell method to overcome the natural complexities in grid generation around arbitrary surface of roughness. The new method combines a non-uniform-grid finite-difference method for discrete grid points near the solid boundary and a shock-fitting method for the treatment of the bow shock. The non-uniform-grid finite-difference formulas are expressed in a general explicit form so that they can be applied to different multi-dimensional problems without any modification. The computational accuracy of new algorithms of up to O(h⁴) are tested on several one- and two-dimensional elliptic equations in irregular domains. In addition, the new method is applied to the simulation of the receptivity process of a Mach 5.92 flow over a flat plate under the combined effect of an isolated surface roughness element and surface blow and suction. A good agreement is found between the unsteady flow results and those obtained by a Linear Stability Theory (LST).     

Deforming composite grids for solving fluid structure problems

We describe a mixed Eulerian–Lagrangian approach for solving fluid–structure interaction (FSI) problems. The technique, which uses deforming composite grids (DCG), is applied to FSI problems that couple high speed compressible flow with elastic solids. The fluid and solid domains are discretized with composite overlapping grids. Curvilinear grids are aligned with each interface and these grids deform as the interface evolves. The majority of grid points in the fluid domain generally belong to background Cartesian grids which do not move during a simulation. The FSI-DCG approach allows large displacements of the interfaces while retaining high quality grids. Efficiency is obtained through the use of structured grids and Cartesian grids. The governing equations in the fluid and solid domains are evolved in a partitioned approach. We solve the compressible Euler equations in the fluid domains using a high-order Godunov finite-volume scheme. We solve the linear elastodynamic equations in the solid domains using a second-order upwind scheme. We develop interface approximations based on the solution of a fluid–solid Riemann problem that results in a stable scheme even for the difficult case of light solids coupled to heavy fluids. The FSI-DCG approach is verified for three problems with known solutions, an elastic-piston problem, the superseismic shock problem and a deforming diffuser. In addition, a self convergence study is performed for an elastic shock hitting a fluid filled cavity. The overall FSI-DCG scheme is shown to be second-order accurate in the max-norm for smooth solutions, and robust and stable for problems with discontinuous solutions for a wide range of constitutive parameters.     

A rapid boundary perturbation algorithm for scattering by families of rough surfaces

In this paper we describe a novel algorithm for the computation of scattering returns by families of rough surfaces. This algorithm makes explicit use of the fact that some scattering profiles of engineering interest (e.g., traveling ocean waves) come in branches parameterized analytically by a bifurcation quantity. Our approach delivers recursions which not only can be implemented to yield a rapid, robust and high-order numerical scheme, but also give a new proof of analyticity of scattering quantities with respect to the bifurcation parameter of the surface family. The real advantage of this new approach is that it computes, in one step, the scattered field for all possible members of the family of surfaces. By contrast, other state-of-the-art schemes must restart when the returns from a new surface are desired, so that the cost of our new approach is greatly advantaged when the number of samples of the family reaches even modest values. Numerical results which verify the accuracy of our approach and demonstrate their utility in computing grating efficiencies scattered by traveling surface ocean waves are presented.     

A unified approach for least-squares surface fitting

This paper presents a novel approach for least-squares fitting of complex surface to measured 3D coordinate points by adjusting its location and/or shape. For a point expressed in the machine reference frame and a deformable smooth surface represented in its own model frame, a signed point-to-surface distance function is defined, and its increment with respect to the differential motion and differential deformation of the surface is derived. On this basis, localization, surface reconstruction and geometric variation characterization are formulated as a unified nonlinear least-squares problem defined on the product space SE(3) × R ^m . By using Levenberg-Marquardt method, a sequential approximation surface fitting algorithm is developed. It has the advantages of implementational simplicity, computational efficiency and robustness. Applications confirm the validity of the proposed approach.     

The surface inhomogeneity of ion-exchange membranes by SEM and AFM data

The surface structure (morphology and roughness) of different types of ion-exchange membranes is investigated via scanning electron microscopy and atomic-force microscopy. It is established that the distribution of ion-exchange regions on the surface of heterogeneous membranes exhibits a complex stochastic character; the sizes of these regions are 5–30 μm. The sizes of irregularities on the surface of homogeneous membranes do not exceed 1 μm. After the samples are subjected to conditioning and the effect of current and temperature, the sizes and portions of the conducting regions of the surface thereof become larger than those of commercial samples by 10–20%. The dependence of the surface microrelief on the membrane type and external factors (chemical conditioning, current, and temperature) is revealed for the first time. The minimum roughness factor corresponds to the surfaces of artificially homogenized and commercial samples of heterogeneous membranes, and the maximum quantities f _r are obtained for membranes with pronounced surface reliefs after current and temperature effects.     

Wavelet frame based scene reconstruction from range data

How to reconstruct the scene (a visible surface) from a set of scattered, noisy and possibly sparse range data is a challenging problem in robotic navigation and computer graphics. As most real scenes can be modeled by piecewise smooth surfaces, traditional surface fitting techniques (e.g. smoothing spline) generally can not preserve sharp discontinuities of surfaces. Based on sparse approximation of piecewise smooth functions in frame domain, we propose a new tight frame based formulation for reconstructing a piecewise smooth surface from a sparse range data set, which is robust to both additive noise and outliers. Furthermore, the resulting minimization problem from our formulation can be efficiently solved by the split Bregman method [1], [2]. The numerical experiments show that the proposed approach is capable of reconstructing a piecewise smooth surface with sharp edges from sparse range data corrupted with noise and outliers.     

Numerical simulation methods for rough surface scattering

Abstract

Numerical methods are of great importance in the study of electromagnetic scattering from random rough surfaces. This review provides an overview of rough surface scattering and application areas of current interest, and surveys research in numerical simulation methods for both one- and two-dimensional surfaces. Approaches considered include numerical methods based on analytical scattering approximations, differential equation methods and surface integral equation methods. Emphasis is placed on recent advances such as rapidly converging iterative solvers for rough surface problems and fast methods for increasing the computational efficiency of integral equation solvers.     

Phantom investigation of phase-inversion-based dual-frequency excitation imaging for improved contrast display

Objective

The goal of this work is to examine the effects of pulse-inversion (PI) technique in combination with dual-frequency (DF) excitation method to separate the high-order nonlinear responses from microbubble contrast agents for improvement of image contrast. DF excitation method has been previously developed to induce the low-frequency ultrasound nonlinear responses from bubbles by using the composition of two high-frequency sinusoids (f₁ and f₂).

Motivation

Although the simple filtering was conventionally utilized to provide signal separation, the PI approach is better in the sense that it minimizes the mutual interferences among these high-order nonlinear responses in the presence of spectral overlap. The novelty of the work is that, in addition to the common PI summation, the PI subtraction was also applied in DF excitation method.

Methods

DF excitation pulses having an envelope frequency of 3 MHz (i.e., f₁ = 8.5 MHz and f₂ = 11.5 MHz) with pulse lengths of 3-10 μs and the pressure amplitudes from 0.5 to 1.5 MPa were used to interrogate the nonlinear responses of SonoVue™ microbubbles in the phantom experiments. The high-order nonlinear responses in the DF excitation were extracted for contrast imaging using PI summation for even-order nonlinear components or PI subtraction for odd-order nonlinear ones.

Results

Our results indicated that, as compared to the conventional filtering technique, the PI processing effectively increases the contrast-to-tissue ratio (CTR) of the third-order nonlinear response at 5.5 MHz and the fourth-order nonlinear response at 6 MHz by 2-5 dB. For these high-order nonlinear components, the CTR increase varies with the transmission pressures from 0.5 to 1.5 MPa due to the microbubbles’ displacement induced by the radiation force of DF excitation.

Conclusions

For DF excitation technique, the PI processing can help to extract either the odd-order or the even-order nonlinear components for higher CTR estimates.     

高精度高次非球面最接近球面计算方法

分析了高次非球面与其加工用最接近球面之间的几何关系特点,提出了一种基于1维搜索的高精度高次非球面最接近球面计算方法。该算法可以计算二次或高次凹（凸）非球面的加工用最接近球面半径、球心位置及非球面度。通过计算实例与现有计算最接近球面的方法相比,该算法在计算高次非球面时将最大非球面度从500.8 m减小到30.0 m,在计算二次非球面时计算结果与精确公式法得到的结果一致。计算实例表明该方法计算高次非球面时得到的最接近球面更优、计算精度更高,且适用于任意次非球面最接近球面的精确计算。     

The effect of different treatments on fractal forms of gallium arsenide surface

The surface of epitaxial gallium arsenide (as-grown and subjected to standard chemical treatments for producing metal-GaAs electrical contacts) has been studied by atomic force microscopy. It was shown that the surface relief can be characterized by fractal relief in the local approximation. As-grown epitaxial n-GaAs layers have lower values of the mean relief irregularity and roughness and reveal high surface homogeneity. The most marked relief changes are observed due to plasmachemical deposition of silicon dioxide and its removal in a buffered etch and deionized water rinse. The following finishing treatments make the surface more smooth and homogeneous, especially in ammonia-water mixture, which gives almost an ideal (Gaussian) distribution of irregularities and phase contrast in most cases. The fractal dimensionality of the studied surfaces changes in the relatively narrow limits D _f = 2.5−2.8. The values of the local approximation limit L determining the boundary of the applicability of the fractal approach when describing the surfaces change from 0.039 to 10.99 μm depending on the kind of treatment.     

Wave guiding properties of ribbed surface waveguides in three frequency domains

A longitudinally structured surface waveguide is examined and the effect of the structuring on the wave guiding properties is explored. The waveguide structure is examined in three distinct wavelength regions and the results are discussed and compared. The Frequency domains are the microwave region around f = 400 MHz, the terahertz region around f = 400 GHz and the optical region around f = 400 THz. Special emphasis is placed on regarding the opposing behaviour of the wave attenuation versus the field extension.     

Photobleaching induced damage of biomolecules: Streptavidin as ‘bio’-photoresist

2-dimensional patterning of surfaces in order to address biological questions is of central interest for the investigation of cell–surface interactions and biosensing.Here we report on an approach which makes use of photobleaching of fluorescently-labelled surface-immobilized streptavidin as ‘bio’-photoresist in order to locally in situ interact with surfaces. Reactive oxygen created during the photobleaching process is likely to be involved in this novel surface manipulation process and its effect on surface-immobilized moieties was investigated by Optical Waveguide Lightmode Spectroscopy (OWLS) and on pre-patterned surfaces using fluorescent microscopy. The creation of bio-patterns as an application of this effect was demonstrated by locally exchanging streptavidin and phospholipid vesicles.This novel surface modification method is interesting from the engineering point of view since no harsh conditions are required and therefore the ‘bio’-photoresist streptavidin allows for the manipulation of surfaces in situ in aqueous solutions e.g., for cell studies.     

Geometric conservation law and applications to high-order finite difference schemes with stationary grids

The geometric conservation law (GCL) includes the volume conservation law (VCL) and the surface conservation law (SCL). Though the VCL is widely discussed for time-depending grids, in the cases of stationary grids the SCL also works as a very important role for high-order accurate numerical simulations. The SCL is usually not satisfied on discretized grid meshes because of discretization errors, and the violation of the SCL can lead to numerical instabilities especially when high-order schemes are applied. In order to fulfill the SCL in high-order finite difference schemes, a conservative metric method (CMM) is presented. This method is achieved by computing grid metric derivatives through a conservative form with the same scheme applied for fluxes. The CMM is proven to be a sufficient condition for the SCL, and can ensure the SCL for interior schemes as well as boundary and near boundary schemes. Though the first-level difference operators δ₃ have no effects on the SCL, no extra errors can be introduced as δ₃ = δ₂. The generally used high-order finite difference schemes are categorized as central schemes (CS) and upwind schemes (UPW) based on the difference operator δ₁ which are used to solve the governing equations. The CMM can be applied to CS and is difficult to be satisfied by UPW. Thus, it is critical to select the difference operator δ₁ to reduce the SCL-related errors. Numerical tests based on WCNS-E-5 show that the SCL plays a very important role in ensuring free-stream conservation, suppressing numerical oscillations, and enhancing the robustness of the high-order scheme in complex grids.     

Finite differences for coarse azimuthal discretization and for reduction of effective resolution near origin of cylindrical flow equations

In this paper, the errors generated by the computation of derivatives in the azimuthal direction θ when flow equations are solved in cylindrical coordinates using finite differences are investigated. They might be large for coarse discretizations even using high-order schemes, which led us to design explicit finite differences specially for 8, 16, 32 and 64 points per circle. These schemes are shown to improve accuracy with respect to standard finite differences, and to provide solutions for a two-dimensional propagation problem similar to those obtained using Fourier spectral methods in the direction θ. A method is then presented to alleviate the time-step limitation resulting from explicit time integration near cylindrical origin, when finite differences are used. It consists in calculating azimuthal derivatives at coarser resolutions than permitted by the grid, in the same way as usually done using spectral methods. In practice, a series of doublings of the effective discretization in θ is implemented. Thus simulations can for instance be performed on a grid containing n_θ = 256 points with a time step 32 times larger, with an accuracy comparable to that achieved in corresponding simulations involving Fourier spectral methods in the direction θ.