Spherical Domain Research Articles

Planar motion controller design for a modular mechatronic device with heading compensation

MechaCells are designed as closed, scalable and modular semi-autonomous devices that can be used alone or part of a pack. In this paper, we discuss a locomotion system that uses the reaction force produced by a rotating unbalance that moves in a spherical domain with a steering mechanism. In order to produce the precise motion capability, a multi-loop controller is developed. This controller uses a friction compensation algorithm based on the mathematical model of the locomotion system. To improve the accuracy of tracking, conventional LuGre friction estimation model is extended for rapid directional changes of the MechaCell during planar motion. The linear and rotational acceleration of the device is also included in controller calculations since it affects the locomotion force generated by the unbalanced mass. The resulting control system is validated both with simulations and experiments and the effectiveness of the extended model and the controller is verified. Our results show significant improvement when a detailed friction compensation observer is used in the controller that includes the effect of sudden steering changes for precise path following.

A non-perturbative method for gravitational potential calculations within heterogeneous and aspherical planets

SUMMARY We present a numerically exact method for calculating the internal and external gravitational potential of aspherical and heterogeneous planets. Our approach is based on the transformation of Poisson’s equation into an equivalent equation posed on a spherical computational domain. This new problem is solved in an efficient iterative manner based on a hybrid pseudospectral/spectral element discretization. The main advantage of our method is that its computational cost reflects the planet’s geometric and structural complexity, being in many situations only marginally more expensive than boundary perturbation theory. Several numerical examples are presented to illustrate the method’s efficacy and potential range of applications.

Critical points of solutions for the mean curvature equation in strictly convex and nonconvex domains

In this paper, we mainly investigate the set of critical points associated to solutions of mean curvature equation with zero Dirichlet boundary condition in a strictly convex domain and a nonconvex domain respectively. Firstly, we deduce that mean curvature equation has exactly one nondegenerate critical point in a smooth, bounded and strictly convex domain of $\mathbb{R}^{n}(n\geq2)$. Secondly, we study the geometric structure about the critical set $K$ of solutions $u$ for the constant mean curvature equation in a concentric (respectively an eccentric) spherical annulus domain of $\mathbb{R}^{n}(n\geq3)$, and deduce that $K$ exists (respectively does not exist) a rotationally symmetric critical closed surface $S$. In fact, in an eccentric spherical annulus domain, $K$ is made up of finitely many isolated critical points ($p_1,p_2,\cdots,p_l$) on an axis and finitely many rotationally symmetric critical Jordan curves ($C_1,C_2,\cdots,C_k$) with respect to an axis.

Stable recurrent calculation of isotropic Gaussian filter coefficients

Filtering in the spherical harmonic domain has become a common practice for the post-processing of temporal gravity field solutions derived from the Gravity Recovery and Climate Experiment (GRACE) satellite mission. The most frequently used filtering technique to suppress high-frequency noise in GRACE data is the isotropic Gaussian averaging filter. The implementation of the isotropic Gaussian filter in the spherical harmonic domain requires the calculation of filter coefficients, which is performed using either a forward recurrence relation, or numerical integration. Both methods show limitations, namely, the forward recurrence relation is numerically unstable and the numerical integration suffers from low accuracy. The objective of this study is to investigate these open problems more thoroughly and to provide alternative expressions for the calculation of isotropic Gaussian filter coefficients. We test three methods (i.e., backward recurrence, continued fraction and sweep algorithms) for the stable evaluation of three-term recurrence relations and examine their properties. All methods successfully solve the instability issues of the forward recurrence relation. They are also more accurate and more computationally efficient than numerical integration.

Weakly nonlinear mode interactions in spherical Rayleigh–Bénard convection

In an annular spherical domain with separation $d$, the onset of convective motion occurs at a critical Rayleigh number $Ra=Ra_{c}$. Solving the axisymmetric linear stability problem shows that degenerate points $(d=d_{c},Ra_{c})$ exist where two modes simultaneously become unstable. Considering the weakly nonlinear evolution of these two modes, it is found that spatial resonances play a crucial role in determining the preferred convection pattern for neighbouring modes $(\ell ,\ell \pm 1)$ and non-neighbouring even modes $(\ell ,\ell \pm 2)$. Deriving coupled amplitude equations relevant to all degeneracies, we outline the possible solutions and the influence of changes in $d,Ra$ and Prandtl number $Pr$. Using direct numerical simulation (DNS) to verify all results, time periodic solutions are also outlined for small $Pr$. The $2:1$ periodic signature observed to be general for oscillations in a spherical annulus is explained using the structure of the equations. The relevance of all solutions presented is determined by computing their stability with respect to non-axisymmetric perturbations at large $Pr$.

Local representation and construction of Beltrami fields II.solenoidal Beltrami fields and ideal MHD equilibria

Object of the present paper is the local theory of solution for steady ideal Euler flows and ideal MHD equilibria. The present analysis relies on the Lie-Darboux theorem of differential geometry and the local theory of representation and construction of Beltrami fields developed in [1]. A theorem for the construction of harmonic orthogonal coordinates is proved. Using such coordinates families of solenoidal Beltrami fields with different topologies are obtained in analytic form. Existence of global solenoidal Beltrami fields satisfying prescribed boundary conditions while preserving the local representation is considered. It is shown that only singular solutions are admissible, an explicit example is given in a spherical domain, and a theorem on existence of singular solutions is proven. Local conditions for existence of solutions, and local representation theorems are derived for generalized Beltrami fields, ideal MHD equilibria, and general steady ideal Euler flows. The theory is applied to construct analytic examples.

Spherical Domain Rate-Distortion Optimization for Omnidirectional Video Coding

Efficient compression of omnidirectional video is important for emerging virtual reality applications. To compress this kind of video, each frame is projected to a 2D plane [e.g., equirectangular projection (ERP) map] first, adapting to the input format of existing video coding systems. At the display side, an inverse projection is applied to the reconstructed video to restore signals in spherical domain. Such a projection, however, makes presentation and encoding in different domains. Thus, an encoder agnostic to the projection performs inefficiently. In this paper, we analyze how a projection influences the distortion measurements in different domains. Based on the analysis, we propose a scheme to optimize the encoding process based on signals’ distortion in spherical domain. With the proposed optimization, an average 4.31% (up to 9.67%) luma BD-rate reduction is achieved for ERP in random access configuration. The corresponding bit saving is averagely 10.84% (up to 34.44%) when considering the viewing field being $\pi /2$ . The proposed method also benefits other projections and viewport settings, with a marginal complexity increase.

Directional Equalization of Sparse Head-Related Transfer Function Sets for Spatial Upsampling

Acquiring decent full-spherical sets of head-related transfer functions HRTFs based on a small number of measurements is highly desirable. For spatial upsampling, HRTF interpolation in the spatially continuous spherical harmonics SH domain is a common approach. However, the number of measured HRTFs limits the assessable SH order, resulting in order-limited HRTFs when transformed to the SH domain. Thus, the SH representation of sparse HRTF sets shows restricted spatial resolution and suffers from order-limitation errors. We present a method that reduces these errors by a directional equalization prior to the SH transform. This is done by a spectral division of each HRTF with a corresponding directional rigid sphere transfer function. The processing removes direction-dependent temporal and spectral components and, therefore, significantly reduces the spatial complexity of the HRTF set, allowing for an enhanced interpolation of HRTFs at reduced SH orders. Spatial upsampling is achieved by an inverse SH transform on an arbitrary dense sampling grid. A subsequent de-equalization by a spectral multiplication with the rigid sphere transfer function recovers the energy in higher spatial orders that was not inherent in the sparse HRTF set. For evaluation, HRTFs were calculated for various limited orders from sparse datasets and compared to a reference. The results show that the proposed method clearly outperforms common SH interpolation of HRTF spectra regarding the overall spectral and temporal structure as well as modeled localization performance.

Two temperature fractional order thermoelasticity theory in a spherical domain

This article is an application of fractional thermoelasticity in association with two-temperature theory. The fractional heat conduction model has been proposed to investigate the thermal variations within the bounded spherical region. The corresponding heat conduction equation has been derived in the context of the generalized two-temperature theory of fractional thermoelasticity. The analytical solutions of thermal variations have been obtained in the Laplace domain, which are inverted using the Gaver–Stehfest algorithm in the time domain. Kuznetsov convergence criterion has been discussed for the bounded variations and stability of the problem. The delay time translations used in the heat flux vector and the temperature gradient result in the finite speed of thermal wave propagation. As a special case of time fractional derivative, the classical and generalized thermoelasticity theories have been recovered.

Microstructure and Transport Analyses in Polymer Electrolyte Membrane Fuel Cells: Comparing Modeling Approaches

Despite the recent entrance of proton exchange membrane fuel cells (PEMFC) into the automotive industry, performance under low platinum loading is still hampered by prohibitive and poorly understood transport losses inside the heterogeneous catalyst layer (CL). Predictive models do not yet sufficiently capture the transport of oxygen, water, and protons in the CL ionomer phase, due to the complexity of the CL microstructure. These transport rates are highly dependent on the substrate, film thickness, and relative humidity. Confinement effects limit water uptake and transport rates below roughly 100 nm thickness, and substrate effects can induce ionomer structures (i.e. lamellae) that further impact transport rates. Implementing these complexities into computational models allows for the microstructural and transport effects on PEMFC performance to be studied and better understood, especially in devices with low platinum loading. Research presented here examines two pseudo-2d, Newman-type model approaches for examining microstructure and transport in PEMFC functional layers: particle-shell and flooded-agglomerate. Figures 1 and 2 show the differences in model domains for these two approaches, respectively. The particle-shell approach approximates a physically realistic structure by simulating a carbon particle at the center of a thin shell of Nafion. This thin shell is discretized to resolve the complex transport that takes place within it. At each vertical discretization, a single Nafion-wrapped carbon particle is taken to be representative of others at that same functional layer depth. The flooded-agglomerate model also simulates a representative spherical domain for each vertical discretization; however, the domain is made up of a cluster of Nafion-wrapped carbon particles enclosed in an additional layer of electrolyte. The models predict cell polarization to examine the effect of CL microstructure as a function of RH- and thickness-dependent transport properties. Figure 3 shows an example overpotential curve from the flooded-agglomerate model in which interesting behavior can be seen with respect to the Nafion thickness. It shows that the CL overpotentials decrease with decreasing ionomer thickness down to 7 nm. Below 7 nm thickness, decreased proton conductivity limits transport, leading to increasing overpotentials at higher currents. The trends indicate that the balance between proton and oxygen transport rates in the CL Nafion can lead to possible trade-offs in optimizing CL design. These and other results will be presented and discussed, to demonstrate the coupled impacts of Nafion thickness and transport on PEMFC performance, and the impact of modeling approach on CL microstructure design. Keywords – Nafion, Transport, PEM, Fuel Cell, Modeling, Electrochemistry, Flooded Agglomerate Figure 1

Evading Derrick’s theorem in curved space: Static metastable spherical domain wall

A recent analysis by one of the authors\cite{Perivolaropoulos:2018cgr} has pointed out that Derrick's theorem can be evaded in curved space. Here we extend that analysis by demonstrating the existence of a static metastable solution in a wide class of metrics that include a Schwarzschild-Rindler-AntideSitter spacetime (Grumiller metric) defined as $ds^2= f(r) dt^2 - f(r)^{-1} dr^2 - r^2 (d\theta^2 +\sin^2\theta d\phi^2)$ with $f(r)=1-\frac{2Gm}{r}+2br-\frac{\Lambda}{3} r^2$ ($\Lambda<0\; b<0$). This metric emerges generically as a spherically symmetric vacuum solution in a class of scalar-tensor theories\cite{Grumiller:2010bz} as well as in Weyl conformal gravity\cite{Mannheim:1988dj}. It also emerges in General Relativity (GR) in the presence of a cosmological constant and a proper spherically symmetric perfect fluid. We demonstrate that this metric supports a static spherically symmetric metastable soliton scalar field solution that corresponds to a spherical domain wall. We derive the static solution numerically and identify a range of parameters $m, b, \Lambda$ of the metric for which the spherical wall is metastable. Our result is supported by both a minimization of the scalar field energy functional with proper boundary conditions and by a numerical simulation of the scalar field evolution. The metastable solution is very well approximated as $\phi(r) = Tanh\left[q (r-r_0)\right]$ where $r_0$ is the radius of the metastable wall that depends on the parameters of the metric and $q$ determines the width of the wall. We also find the gravitational effects of the thin spherical wall solution and its backreaction on the background metric that allows its formation. We show that this backreaction does not hinder the metastability of the solution even though it can change the range of parameters that correspond to metastability.

OmniCast: Wireless Pseudo-Analog Transmission for Omnidirectional Video

Wireless virtual reality (VR) applications that provide users extraordinary viewing experience are now drawing great attentions. Transmitting VR video via wireless channel to users’ head-mounted display devices efficiently with low latency is very important for many emerging VR applications. In this paper, we propose a pseudo-analog transmission framework named OmniCast, which provides graceful quality degradation and competitive performance for unpredictably varying wireless channels while featuring low latency, low complexity, and low energy cost. In particular, we analyze the influence of projection between the spherical representation and the 2-D plane representation, and derive a power optimization scheme to minimize the distortion on the sphere. In addition, we develop an approach to measure the efficiency of decorrelation transform in the spherical domain. Based on that, an appropriate transform option can be determined. Experimental results show that the proposed framework improves the transmission efficiency of omnidirectional videos, while achieving elegant quality degradation for channel fluctuation in a wide channel SNR range.

Direction of Arrival Estimation for Reverberant Speech Based on Enhanced Decomposition of the Direct Sound

Direction of arrival (DOA) estimation for speech sources is an important task in audio signal processing. This task becomes a challenge in reverberant environments, which are typical to real scenarios. Several methods of DOA estimation for speech sources have been developed recently, in an attempt to overcome the effect of reverberation. One effective approach aims to identify time-frequency bins in the short time Fourier transform domain that are dominated by the direct sound. This approach was shown to be particularly adequate for spherical arrays, with processing in the spherical harmonics domain. The direct-path dominance (DPD) test, and a method which is based on the directivity of the sound field are recent examples. While these methods seem to perform well, high reverberation conditions may degrade their performance. In this paper, the structure of the spatial correlation matrix is comprehensively studied, showing that under some well-defined conditions, the DOA of the direct sound can be correctly extracted from its dominant eigenvector, even when contaminated by reflections. This new insight leads to the development of a new test, performing an enhanced decomposition of the direct sound (EDS), denoted the DPD-EDS test. The proposed test is compared to previous DPD tests, and to other recently proposed reverberation-robust methods, using computer simulations and an experimental study, demonstrating its potential advantage. The studies include multiple speakers in highly reverberant environments, therefore representing challenging real-life acoustics scenes.

Rock avalanche-debris geometry and implications for rock-avalanche genesis

Rock avalanches have an important impact on the geomorphological evolution of mountain landscapes. Rock avalanche-debris geometry holds important clues related to the processes of the disaster predisposition, occurrence and rapid runout of rock avalanche-debris flows. The morphological features of the debris of the Touzhai rock avalanche deposit in Southwest China are examined as a specific case with the help of three-dimensional laser scanning, polarization microscopy (for polished sections) and field-emission scanning electron microscopy (FESEM). The unexpectedly high roundness is the most impressive morphological characteristic of the Touzhai rock avalanche-debris, particularly that of sand-plus silt-sized particles. The shapes of 12 scanned metric-sized boulders are closer to cubes than to spheres. 90% of the 114 scanned subangular blocks with a mean particle diameter of 12.04 cm and an average specific surface area of 0.225 cm2/g fall into the spherical domain of Zingg's classification diagram according to their flatness ratio and elongation ratio, and the average sphericity of 0.798 further indicates that they are more akin to cubes compared with spheres. The average form factor of the 1201 sand-sized grains is 0.784 with a standard deviation of 0.044, indicating that they are nearly spherical, and the solidity ranging from 0.951 to 0.974 also strongly shows that these particles are close to smooth spheres. Most of the grains in the size range of <0.075 mm are sub-angular except slaty secondary clay minerals of <0.005 mm based on the visual estimation under FESEM. The detailed qualitative and quantitative contrasts between the morphometric characteristics of the Touzhai rock-avalanche debris and those of the fresh basalt fragments demonstrate that the geometry of the former is dominated by the pre-failure predesign in which the basalt, primary and secondary structures, unloading and weathering acted together. Weathering was the direct contributor to its high roundness. The dynamic fragmentation triggering newly generated ruptures during the post-failure motion was less important, which should be the case in some rock avalanches in other regions. The slope pattern of “large creeping main slide mass + lower intact locked segment” generalized according to this study should be applicable to some other rock avalanches worldwide, and can help the pre-event identification of the potential rock-avalanche sites. Our results support the hypothesis of “aerification of debris” to explain the high mobility of rock avalanches. Highly fragmented source rockmass with enough fine grains should be necessary for the occurrence of rock avalanches and long, rapid runout of the rock avalanche-debris flows.

An efficient spectral‐Galerkin method based on a dimension reduction scheme for eigenvalue problems of Schrödinger equations

In this paper, we propose an efficient spectral‐Galerkin method based on a dimension reduction scheme for eigenvalue problems of Schrödinger equations. Firstly, we carry out a truncation from a three‐dimensional unbounded domain to a bounded spherical domain. By using spherical coordinate transformation and spherical harmonic expansion, we transform the original problem into a series of one‐dimensional eigenvalue problem that can be solved effectively. Secondly, we introduce a weighted Sobolev space to treat the singularity in the effective potential. Using the property of orthogonal polynomials in weighted Sobolev space, the error estimate for the approximate eigenvalues and corresponding eigenfunctions are proved. Error estimates show that our numerical method can achieve spectral accuracy for approximate eigenvalues and eigenfunctions. Finally, we give some numerical examples to demonstrate the efficiency of our algorithms and the correctness of the theoretical results.

Spatial sound intensity vectors in spherical harmonic domain.

Sound intensity is a fundamental quantity describing acoustic wave fields and it contains both energy and directivity information. It is used in a variety of applications such as source localization, reproduction, and power measurement. Until now, intensity is defined at a point in space, however given sound propagates over space, knowing its spatial distribution could be more powerful. This paper formulates spatial sound intensity vectors in spherical harmonic domain such that the vectors contain energy and directivity information over continuous spatial regions. These representations are derived with finite sets of closed form coefficients enabling ease of implementation.

Different Types Domains are Present in Complex I from Immature Seeds and of CA Adult Plants in Arabidopsis thaliana.

Mitochondrial Nicotinamide adenine dinucleotide (NADH) dehydrogenase complex is the first complex of the mitochondrial electron transfer chain. In plants and in a variety of eukaryotes except Opisthokonta, complex I (CI) contains an extra spherical domain called carbonic anhydrase (CA) domain. This domain is thought to be composed of trimers of gamma type CA and CA-like subunits. In Arabidopsis, the CA gene family contains five members (CA1, CA2, CA3, CAL1 and CAL2). The CA domain appears to be crucial for CI assembly and is essential for normal embryogenesis. As CA and CA-like proteins are arranged in trimers to form the CA domain, it is possible for the complex to adopt different arrangements that might be tissue-specific or have specialized functions. In this work, we show that the proportion of specific CI changes in a tissue-specific manner. In immature seeds, CI assembly may be indistinctly dependent on CA1, CA2 or CA3. However, in adult plant tissues (or tissues derived from stem cells, as cell cultures), CA2-dependent CI is clearly the most abundant. This difference might account for specific physiological functions. We present evidence suggesting that CA3 does not interact with any other CA family member. As CA3 was found to interact with CI FRO1 (NDUFS4) subunit, which is located in the matrix arm, this suggests a role for CA3 in assembly and stability of CI.

A multi-GPU finite volume solver for magnetohydrodynamics-based solar wind simulations

Magnetohydrodynamic (MHD) simulations in the domain of spherical shell are a crucial and challenging subject in many fields such as geophysics and solar-terrestrial physics, due to the complication of the MHD equations and the specificity of the domain. Besides, due to the real-time requirement, accelerating the heavy computation is proposed in many practical problems, of which the space weather simulation and forecast from the Sun to Earth is a typical case. Considering these factors, we first develop a new, spatially second-order accurate finite volume (FV) solver for three-dimensional (3D) MHD simulations with the multiple time steps strategy, which is based on the six-component grid for spherical shell domain. Then to speed up the simulation, we implement the solver on multiple GPUs with optimizations of CUDA and establish an effective multi-GPU FV solver on the spherical shell domain. A MHD manufactured solution is used to validate the solvers’ spatial accuracy, and to measure their performances. Results show that both solvers have nice scalability, and speedup ratios of 27.7x to 30.06x are obtained on GPUs. Then we utilize them to study the ambient solar wind for Carrington rotation (CR) 2060. The multi-GPU FV solver can not only obtain speedup ratios of about 29.0x, but capture main features of the solar corona and the mapped in-situ solar wind measurements.

Light in power

We present in this paper a generic and parameter-free algorithm to efficiently build a wide variety of optical components, such as mirrors or lenses, that satisfy some light energy constraints. In all of our problems, one is given a collimated or point light source and a desired illumination after reflection or refraction and the goal is to design the geometry of a mirror or lens which transports exactly the light emitted by the source onto the target. We first propose a general framework and show that eight different optical component design problems amount to solving a light energy conservation equation that involves the computation of visibility diagrams. We then show that these diagrams all have the same structure and can be obtained by intersecting a 3D Power diagram with a planar or spherical domain. This allows us to propose an efficient and fully generic algorithm capable to solve these eight optical component design problems. The support of the prescribed target illumination can be a set of directions or a set of points located at a finite distance. Our solutions satisfy design constraints such as convexity or concavity. We show the effectiveness of our algorithm on simulated and fabricated examples.

Mitigating wind noise with a spherical microphone array.

This paper utilizes a rigid spherical microphone array to reduce wind noise. In the experiments conducted, a loudspeaker is used to reproduce the desired sound signal and an axial fan is employed to generate wind noise in an anechoic chamber. The sound signal and wind noise are measured separately with the spherical microphone array and analyzed in the spherical harmonic domain. The wind noise is found to be irregularly distributed in the spherical harmonic domain, distinct from the sound signal which is concentrated in the first few spherical harmonic modes. This difference is utilized to reduce wind noise without degrading the desired sound pressure level (SPL) by use of a low pass filter method in the spherical harmonic domain. Experimental results with both single-tonal and multi-tonal sound signals demonstrate that the proposed method can reduce wind noise by more than 10 dB in the frequency range below 500 Hz. The SPL of the desired sound signal can be extracted from wind noise with an error within 1.0 dB, even when the sound level is 8 dB lower than wind noise.