Boundary Value Problem Approach Research Articles

Connecting the Brazilian Vertical System to the International Height Reference Frame by estimating the vertical datum parameters

One of the challenges in the unification of the global vertical datum is to determine the vertical offset between the local vertical datum and the global vertical datum. Since 2015, the scientific community has been working on the unification of the global vertical datum, following the publication of the resolution for the definition and realization of an International Height Reference System. This paper aims to estimate the vertical offset in Brazil (Imbituba and Santana Data) by combining a regional gravity field model, using the Geodetic Boundary Value Problem approach applying the Remove-Compute-Restore procedure, and GNSS/leveling data. In the Imbituba vertical datum 1271 stations were used and 66 stations were connected to the Santana tide gauge. The results showed that the geopotential value of Imbituba was 62,636,849.87 ± 0.224 m2s-2, and the vertical offset concerning the reference surface (W0) was 0.358 ± 0.023 m. In Santana, the geopotential value was 62,636,836.88 ± 4.108 m2s-2, and the vertical offset was 1.689 ± 0.420 m. The offset between Imbituba and Santana was computed once the physical connection between Brazil's two local vertical data is a challenge. The estimated value was 1.331 ± 0.421 m. This means the Santana vertical datum is 1.331 m higher than the Imbituba vertical datum.

A new very high-order finite-difference method for linear stability analysis and bi-orthogonal decomposition of hypersonic boundary layer flow

Precisely predicting laminar-turbulence transition locations is essential for improvements in hypersonic vehicle design related to flow control and heat protection. Currently state-of-the-art eN prediction method requires the evaluation of discrete normal modes F and S for the growth rate of instability wave. Meanwhile, in receptivity studies, both the discrete and continuous modes, including acoustic, entropy, and vorticity modes, contribute to the generation of the initial disturbance. The purpose of this paper is to introduce a new very high-order numerical method to accurately compute these normal modes with finite-difference on a non-uniform grid. Currently, numerical methods to obtain these normal modes include two major approaches, the boundary value problem approach and the initial value problem approach. The boundary value approach used by Malik (1990) [17] deploys fourth-order finite difference and spectral collocation methods to solve a boundary value problem for linear stability theory (LST). Nonetheless, Malik's presentation only demonstrated the computation of discrete modes, but not the continuous modes essential for conducting modal analysis on receptivity data. To obtain the continuous spectrum for his multimode decomposition framework, Tumin (2007) [16] relies on an initial value approach based on the Runge Kutta scheme with the Gram-Schmidt orthonormalization. However, the initial value approach is a local method that does not give a global evaluation of the eigenvalue spectra of discrete modes. Furthermore, Gram-Schmidt orthonormalization, which can be error-prone in implementation, is required at every step of the integration to minimize the accumulation of numerical errors. To overcome the drawbacks of these two approaches, this paper improves the boundary value approach by introducing a new general very high-order finite difference method for both discrete and continuous modes eigenfunctions. This general high-order finite difference method is based on a non-uniform grid method proposed by Zhong and Tatineni (2003) [22]. Under the finite difference framework, discrete and continuous modes can be obtained by imposing proper freestream asymptotic boundary conditions based on the freestream fundamental solution behavior. This asymptotic boundary condition is used for obtaining both discrete and continuous modes that have both distinct (acoustic) and similar (vorticity and entropy) eigenvalues. Extensive verification of the new method has been carried out by comparing the computed discrete and continuous modes. Subsequently, the discrete and continuous modes obtained with this finite difference method are essential for the bi-orthogonal decomposition, which holds promising potential in obtaining an accurate evaluation of receptivity coefficients. The result of the bi-orthogonal decomposition for a hypersonic boundary layer flow over a flat plate is verified by comparing with existing results. Ultimately, the bi-orthogonal decomposition using the eigenfunctions has been applied to a case of freestream receptivity simulation for an axis-symmetric hypersonic flow over a blunt nose cone with modal contributions computed as coefficients for receptivity analysis.

Analysis of a Relative Offset between the North American and the Global Vertical Datum in Gravity Potential Space

The accurate estimation of the zero-height geopotential level in a local vertical datum (LVD) is critical for linking traditional height reference systems to a global height system. In this paper, we investigate the theoretical and practical challenges involved in determining the offset between the North American vertical datum (NAVD) and the global vertical datum (GVD). Drawing on the classical theory of the vertical system in physical geodesy, we define the vertical datum offset and derive rigorous formulas for its calculation. We examine various factors that affect the determination of the offset, including the global gravitational models (GGMs), geodetic reference system, tide system, tilt error, and omission error. Using terrestrial gravity data and gravity anomalies from multiple GGMs in conjunction with Global Navigation Satellite System (GNSS) and orthometric heights, we estimate the vertical offset between the NAVD and GVD. Our results indicate that the geopotential difference approach and the geodetic boundary value problem (GBVP) approach yield consistent results. When the normal gravity geopotential of the geodetic reference system is selected as the gravity geopotential of the global height datum, the NAVD is approximately 0.04 m higher than the GVD relative to the GRS80 ellipsoid, and 0.97 cm higher than the GVD relative to the WGS84 ellipsoid. When the Gauss–Listing geopotential value is chosen as the gravity geopotential of the global height datum, the NAVD is roughly 1.45 m higher than the GVD relative to the GRS80 ellipsoid, and approximately 0.52 m higher than the GVD relative to the WGS84 ellipsoid.

On the connection of the Ecuadorian Vertical Datum to the IHRS

Abstract In this work, the determination of the discrepancy between the Ecuadorian Vertical Datum (EVD) and the International Height Reference System (IHRS) is presented. The vertical offset was estimated at the EVD based on the fixed geodetic boundary value problem approach. The focus of the experiment was the determination of the anomalous potential in the EVD, which in turn enable determination of the respective geopotential value. Taking a geopotential space-based approach, two estimates of the EVD offset with respect to the IHRS were obtained that amount to −1.51 and −1.61 m2/s2.

An optimal immunotherapeutic treatment of HIV infections to regain the targeted CD4 + T cell count: a boundary value problem approach

An optimal immunotherapeutic treatment of HIV infections to regain the targeted CD4 + T cell count: a boundary value problem approach

Estimation of Geopotential Value W0 for the Geoid and Local Vertical Datum Parameters

Unification of the global vertical datum has been a key problem to be solved for geodesy over a long period, and the main challenge for unifying the global vertical datum system is to determine the geopotential value W0 of the geoid and to calculate the vertical offset between the local vertical datum and the global vertical datum W0. The geopotential value W0 can be calculated using the grid mean sea surface (GMSS) data and the global geopotential model (GGM). In this study, this GMSS data was measured with adjustment methods and 24 years of merged multi-satellite altimetry data. The data of HaiYang-2A (HY-2A) and Jason-3 were first used to calculate W0. The geopotential value W0 was determined to be 62,636,856.82 m2s−2 by combining the EIGEN-6C4 (European Improved Gravity Model of the Earth by New Techniques) and the GMSS data. Then, the geopotential difference approach and geodetic boundary value problem (GBVP) approach were used to determine the vertical datum parameters in this study. To compensate for the omission error of the GGM, this study utilized the remove–compute–restore (RCR) technique and the residual terrain model (RTM)-recovered high-frequency gravity signals. Finally, as a result of the GBVP solution, the geopotential value of the Australian Height Datum (AHD) was 62,636,851.935 m2s−2, and the vertical offset of the AHD relative to the global vertical datum W0 was 0.4885 m. As a result of the geopotential difference approach, the geopotential value of the Chinese Height datum was 62636861.412 m2s−2, and the vertical offset of the Chinese Height datum was −0.4592 m.

Heat and mass transfer analysis above an unsteady infinite porous surface with chemical reaction

The flow of viscous nanofluid with heat and mass transmission above a porous flat surface in the existence of a constant magnetic field is addressed in this article. Brownian motion, Thermophoresis, and viscous dissipation are all taken into account. The motion is described by a system of equations that are transformed into conventional non-linear differential equations by appropriate transformations and then tackled numerically using the boundary value problem approach at MATLAB. The convergence criterion for the solution is 10 − 6 . The found solutions are functions of the problem's physical properties. The effects of different study parameters for velocity, temperature, and concentration profiles are explored through graphs and tables. An increase in the Prandtl number increases the Sherwood and Nusselt number while Brownian motion reduces the Sherwood number. Viscoelastic parameter increases the skin friction and decreases the nusselt number.

Determination of the geopotential value on the permanent GNSS stations in Vietnam based on the Geodetic Boundary Value Problem approach

SUMMARY In the realisation of the International Height Reference System, the determination of the geopotential value and its variations in time plays an important role. In this study, the geodetic boundary value problem approach is applied for direct determination of the gravity potential value using a GOCE global gravity field model enhanced with terrestrial gravity data. This determination is carried out on the Global Navigation Satellite System-Continuously Operating Reference Stations (GNSS-CORS) stations towards the realisation of the International Height Reference System in Vietnam. First, the effects of the GOCE global gravity field model omission error, the indirect bias term on the disturbing potential and the systematic cumulative errors in levelling data are estimated. These errors affect the estimated geopotential value. The results calculated on the GNSS/levelling points show that the effect of the GOCE DIR-R5 (up to degree/order 260) omission error on the offset potential value is quite significant. This effect was eliminated using high-resolution terrestrial gravity data using the remove-compute-restore technique. The indirect bias term on the disturbing potential can be safely neglected by using a GOCE global gravity field model for degrees higher than 60 for this study region. The systematic cumulative errors in levelling data can be modelled and removed using a third-order polynomial model. Then, the mean zero-height gravity potential of the Vietnam local vertical datum is estimated equal to ${\rm{W}}_0^{{\rm{LVD}}}$ = 62 636 846.69 m2 s–2 with standard deviation of 0.70 m2 s–2 based on the proposed approach. Finally, the geodetic boundary value problem approach was used to determine the geopotential on the surface of three GNSS-CORS stations in Vietnam. Based on time-series of the vertical component derived from the GNSS observations as well as InSAR data, temporal variations in geopotential are also estimated on these permanent GNSS stations. The purpose is to monitor deformation of the vertical datum. The results indicate that the geopotential value needs to be monitored and determined with the time-dependent component on the three Vietnamese permanent GNSS stations for a vertical datum. These stations may contribute to increase the density of reference points in the International Terrestrial Reference Frame, which is being researched and implemented by the International Association of Geodesy.

Geometric control using the state-dependent Riccati equation: application to aerial-acrobatic maneuvers

Acrobatic flip is one of the most challenging representatives of aggressive maneuvers to test the performance of an aerial system’s capability or a controller. A variable-pitch rotor quadcopter generates thrust in both vertical directions for the special design of the rotor’s actuation mechanism. This research proposes two possible solutions for the flip: a regulation solution based on the geometric control approach; and tracking a predefined optimal smooth trajectory covering a turnover. The first solution uses a geometric control approach that is immune to singular points since the rotation matrix is integrated on the manifold on . The second solution proposes an optimal trajectory generation for flip maneuver using open-loop optimal control, two-point boundary value problem (TPBVP) approach. Since generated open-loop state information is not applicable without a controller, the state-dependent differential Riccati equation (SDDRE) is chosen for trajectory tracking.

A fast ADI algorithm for nonlinear Poisson equation in heterogeneous dielectric media

Recently, a nonlinear Poisson equation has been introduced to model nonlinear and nonlocal hyperpolarization effects in electrostatic solute-solvent interaction for biomolecular solvation analysis. Due to a strong nonlinearity associated with the heterogeneous dielectric media, this Poisson model is difficult to solve numerically, particularly for large protein systems. A new pseudo-transient continuation approach is proposed in this paper to efficiently and stably solve the nonlinear Poisson equation. A Douglas type alternating direction implicit (ADI) method is developed for solving the pseudo-time dependent Poisson equation. Different approximations to the dielectric profile in heterogeneous media are considered in the standard finite difference discretization. The proposed ADI scheme is validated by considering benchmark examples with exact solutions and by solvation analysis of real biomolecules with various sizes. Numerical results are in good agreement with the theoretical prediction, experimental measurements, and those obtained from the boundary value problem approach. Since the time stability of the proposed ADI scheme can be maintained even using very large time increments, it is efficient for electrostatic analysis involving hyperpolarization effects.

Estimation of vertical datum offset for the South African vertical datum, in relation to the international height reference system

The vertical offset and the geopotential value over South Africa is estimated on the four fundamental benchmarks in relation to the international height reference system (IHRS). It is estimated to obtain discrepancies between the South African local vertical datum (W_P) and the global vertical datum (W_0). A single-point-based geodetic boundary value problem (GBVP) approach was used following Molodensky theory for estimating the height anomalies from the disturbing potential (T_P) using Bruns’s formula. The gravity potential at each tide gauge benchmark (TGBM) in South Africa deviates from the potential of the global reference surface by 0.589,-1.993,-2.593 and 2.154 m2s-2 for Cape Town, Port Elizabeth, East London and Durban, respectively. The corresponding vertical datum offsets between the international height reference system and the four fundamental benchmarks over South Africa are 6.013, -20.347, -26.478, and 21.996 cm for Cape Town, Port Elizabeth, East London and Durban, respectively. These offsets can be used for the unification of the South African vertical datum at the four tide gauge benchmarks in a manner that is consistent to the international height reference system.

Estimation of Vertical Datum Parameters Using the GBVP Approach Based on the Combined Global Geopotential Models

Unification of the global vertical datum has been a key problem to be solved for geodesy over a long period, and the main challenge for a unified vertical datum system is to determine the vertical offset between the local vertical datum and the global vertical datum. For this purpose, the geodetic boundary value problem (GBVP) approach based on the remove-compute-restore (RCR) technique is used to determine the vertical datum parameters in this paper. In the RCR technique, a global geopotential model (GGM) is required to remove and restore the long wavelengths of the gravity field. The satellite missions of the GRACE (Gravity Recovery and Climate Experiment) and GOCE (Gravity field and steady-state Ocean Circulation Exploration) offer high accuracy medium–long gravity filed information, but GRACE/GOCE-based GGMs are restricted to medium–long wavelengths because the maximum degree of their spherical harmonic representation is limited, which is known as an omission error. To compensate for the omission error of GRACE/GOCE-based GGM, a weighting method is used to determine the combined GGM by combining the high-resolution EGM2008 model (Earth Gravitational Model 2008) and GRACE/GOCE-based GGM to effectively bridge the spectral gap between satellite and terrestrial data. An additional consideration for the high-frequency gravity signals is induced by the topography, and the residual terrain model (RTM) is used to recover the omission errors effect of the combined GGM. In addition, to facilitate practical implementation of the GBVP approach, the effects of the indirect bias term, the spectral accuracy of the GGM, and the systematic levelling errors and distortions in estimations of the vertical datum parameters are investigated in this study. Finally, as a result of the GBVP solution based on the combined DIR_R6/EGM2008 model, RTM, and residual gravity, the geopotential values of the North American Vertical Datum of 1988 (NAVD88), the Australian Height Datum (AHD), and the Hong Kong Principal Datum (HKPD) are estimated to be equal to 62636861.31 ± 0.96, 62653852.60 ± 0.95 and 62636860.55 ± 0.29 m2s−2, respectively. The vertical offsets of NAVD88, AHD, and HKPD with respect to the global geoid are estimated as −0.809 ± 0.090, 0.082 ± 0.093, and −0.731 ± 0.030 m, respectively.

Improved tracklet association for space objects using short-arc optical measurements

Initial orbit determination (IOD) for space objects is challenging, especially in the case where only optical observations, i.e. angles-only observations, are available and the optical observing arcs are very short (i.e. the too-short arc (TSA) problem). One approach to address the TSA problem is to associate several short-arc tracklets to targets across varying time intervals. In order to achieve better association and run-time performance, this study proposes an improvement to the traditional initial value problem (IVP) solution that determines the association by searching for the global minimum of a new loss function defined in a nonsingular canonical space. The improved IVP method was validated using optical data of space objects at different altitudes collected from the Mount Stromlo Observatory and compared with traditional IVP and another popular tracklet association method: the boundary value problem (BVP) approach. Results illustrate that the improved IVP method is superior to IVP and BVP in terms of association performance, and it also achieves good run-time performance. In addition, traditional methods suffer the drawback of incorrectly associating tracklets from different objects in the same constellation. A new approach dubbed the common ellipse method is presented to address this issue. The common ellipse method is tested with 86 Iridium constellation tracklets, and results show that it significantly improves the true negative rate for the tested scenario.

Vertical datum unification for the International Height Reference System (IHRS)

The International Association of Geodesy released in July 2015 a resolution for the definition and realisation of an International Height Reference System (IHRS). According to this resolution, the IHRS coordinates are potential differences referring to the equipotential surface of the Earth's gravity field realised by the conventional value W0 = 62 636 853.4 m2s−2. A main component of the IHRS realisation is the integration of the existing height systems into the global one; i.e. existing vertical coordinates should be referred to one and the same reference level realised by the conventional W0. This procedure is known as vertical datum unification and its main result are the vertical datum parameters, i.e., the potential differences between the local and the global reference levels. In this paper, we rigorously derive the observation equations for the vertical datum unification in terms of potential quantities based on the geodetic boundary value problem (GBVP) approach. Those observation equations are then empirically evaluated for the vertical datum unification of the North American and South American height systems. In the first case, simulations performed in North America provide numerical estimates about the impact of omission errors and direct and indirect effects on the vertical datum parameters. In the second case, a combination of local geopotential numbers, ITRF coordinates, satellite altimetry observations, tide gauge registrations and high-resolution gravity field models is performed to estimate the level differences between the South American height systems and the global level W0. Results show that indirect effects vanish when a satellite-only gravity field model with a degree higher than n ≥ 180 is used for the solution of the GBVP. However, the component derived from satellite-only global gravity models has to be refined with terrestrial gravity data to minimise the omission error and its effect on the vertical datum parameter estimation. The empirical evaluations demonstrate that the vertical datum unification should be based on geodetic stations of highest quality and standardised geodetic data; for example, geometric coordinates should refer to the same ITRF and be given in the same tide system and reference epoch as the geopotential numbers and gravity field model. After a standardisation of the input data used in the unification of the South American height systems and a rigorous error propagation analysis, we demonstrate that the vertical datum parameters can be estimated with accuracy better than ± 5 cm in well-surveyed regions and some decimetres (± 40 cm) in sparsely surveyed regions. This paper concludes with detailed guidelines for the appropriate data treatment when the integration of a local vertical datum into the IHRS is desired. These guidelines may be applicable in any region of the world.

FEM × DEM: a new efficient multi-scale approach for geotechnical problems with strain localization

The paper presents a multi-scale modeling of Boundary Value Problem (BVP) approach involving cohesive-frictional granular materials in the FEM × DEM multi-scale framework. On the DEM side, a 3D model is defined based on the interactions of spherical particles. This DEM model is built through a numerical homogenization process applied to a Volume Element (VE). It is then paired with a Finite Element code. Using this numerical tool that combines two scales within the same framework, we conducted simulations of biaxial and pressuremeter tests on a cohesive-frictional granular medium. In these cases, it is known that strain localization does occur at the macroscopic level, but since FEMs suffer from severe mesh dependency as soon as shear band starts to develop, the second gradient regularization technique has been used. As a consequence, the objectivity of the computation with respect to mesh dependency is restored.

Optimal Push Recovery for Periodic Walking Motions

We study how humans performing periodic walking motions react to strong perturbations that are applied in form of pushes from behind. We propose a computational method that allows to generate optimal recovery motions based on a dynamic multi-body model of the human walking process. The assumption is that humans in such a situation would aim to pursue the walking motions while at the same time limiting the effort for the motion. For a given position and velocity in the middle of a periodic walking motion and a given pushing force profile, magnitude and contact point we determine the motion of the human model that minimizes a combined criterion joint torques and a deviation from periodicity. The recovery horizon considered in the multiphase optimal control problem is one step. The optimal control problem is solved using a direct boundary value problem approach based on multiple shooting. We present resulting optimal recovery motions for pushing forces between 150 N and 600 N which all look realistic. The proposed method has potential applications in the computation of push recovery motions for humanoid robots or for elderly humans with physical assistive devices, in each case applied to the respective model.

Product innovation with lumpy investment

The paper provides a framework that enables us to analyze the important topic of capital accumulation under technological progress. We describe an algorithm to solve Impulse Control problems, based on a (multipoint) boundary value problem approach. Investment takes place in lumps and we determine the optimal timing of technology adoptions as well as the size of the corresponding investments. Our numerical approach led to some guidelines for new technology investments. First, we find that investments are larger and occur in a later stadium when more of the old capital stock needs to be scrapped. Moreover, we obtain that the size of the firm’s investments increase when the technology produces more profitable products. We see that the firm in the beginning of the planning period adopts new technologies faster as time proceeds, but later on the opposite happens. Furthermore, we find that the firm does not invest such that marginal profit is zero, but instead marginal profit is negative.

The GBVP approach for vertical datum unification: recent results in North America

Two levelling-based vertical datums have been used in North America, namely CGVD28 in Canada and NAVD88 in the USA and Mexico. Although the two datums will be replaced by a common and continent-wide vertical datum in a few years, their connection and unification are of great interest to the scientific and user communities. In this paper, the geodetic boundary value problem (GBVP) approach is studied as a rigorous method for connecting two or more vertical datums through computed datum offsets from a global equipotential surface defined by a GOCE-based geoid. The so-called indirect bias term, the effect of the GOCE geoid omission error, the effect of the systematic levelling datum errors and distortions, and the effect of the geodetic data errors on the datum unification are four important factors affecting the practical implementation of this approach. These factors are investigated numerically using the GNSS-levelling and tide gauge stations in Canada, the USA, Alaska, and Mexico. The results show that the indirect bias term can be omitted if a GOCE-based global geopotential model is used in gravimetric geoid computations. The omission of the indirect bias term simplifies the linear system of equations for the estimation of the datum offset(s). Because of the existing systematic levelling errors and distortions in the Canadian and US levelling networks, the datum offsets are investigated in eight smaller regions along the Canadian and US coastal areas. Using GNSS-levelling stations in the US coastal regions, the mean datum offset can be estimated with a 1 cm standard deviation if the GOCE geoid omission error is taken into account by means of the local gravity and topographic information. In the Canadian Atlantic and Pacific regions, the datum offsets can be estimated with 2.3 and 3.5 cm standard deviation, respectively, using GNSS-levelling stations. However, due to the low number of tide gauge stations, the standard deviation of the CGVD28 and NAVD88 datum offsets can reach one decimetre in the Pacific regions. With the available GNSS-levelling stations in Alaska and Mexico, the NAVD88 datum offset can be estimated with a standard deviation below 3 cm. The numerical investigations of this study provide, for the first time, the datum offsets between North American vertical datums and their associated standard deviations with which the offsets can be estimated. The results of this study demonstrate the importance of the aforementioned four factors in the practical implementation of the GBVP approach for the unification of the levelling-based vertical datums.

On a functional equation arising from two processors with coupled inputs

During the last few decades, a certain interesting class of functional equations arises when obtaining the generating functions of many system distributions. Such a class of equations has numerous applications in many modern disciplines like wireless networks and communications. This paper has been motivated by an issue considered by Paul E. Wright in [Advances in applied probability, (1992), 986 ô€€€ 1007]. The functional equation obtained there has been solved using elliptic functions and analytic continuation, which in turn lead to the determination of the main unknown. Unfortunately that solution seems to be a bit too general with many technical assumptions. In this paper on one hand, we introduce a solution in the symmetric case using boundary value problem approach. On the other hand, we investigate the potential singularities of the unknowns of the functional equation giving one possible application, and we compute some expectation of interest using the corresponding generating function.

Vibration analysis and robust control of highly deformable beams in a heavy pinched loop configuration

A heavy pinched loop is formed by bringing and clamping the two ends of a highly deformable slender beam, elastica. A collocation solution technique is implemented for studying the formation statically and dynamically, i.e. small vibrations around the large deformed static solutions, and the earlier work using a shooting method is validated. A new and clear Galerkin formulation capable of modelling damping is established for finding transients, and a new theoretical multi-point boundary value problem approach is used for numerically obtaining the frequency response function. Lastly, the obtained dynamic model is used for active vibration control, wherein a controller is designed using H ∞ algorithm for active damping in a heavy pinched loop for two simplified cases, and the simulated results are shown.