Incorporating Boundary Conditions Research Articles

Fast flow field prediction based on E(2)-equivariant steerable convolutional neural networks

In the field of flow field reconstruction, traditional deep learning models predominantly rely on standard convolutions, but their predictive accuracy remains limited. To address this issue, we explore the potential of E(2)-equivariant convolutions to enhance the predictive accuracy of deep learning models for fast flow field prediction. Unlike conventional convolutions, E(2)-equivariant convolutions offer a richer representation capability by better capturing geometric and structural information. Our neural network integrates an attention mechanism that leverages the signed distance function (SDF) to encode geometric details and an indicator matrix to incorporate boundary conditions. The model predicts velocity and pressure fields as outputs. We conducted experiments specifically targeting non-uniform steady laminar flows, and the results show a 16.1% reduction in overall error compared to models based on traditional convolutions while maintaining high efficiency. These findings indicate that E(2)-equivariant convolution, coupled with an attention mechanism, significantly improves flow field prediction by focusing on critical information and better representing complex geometries.

Minimizers for the de Gennes–Cahn–Hilliard energy under strong anchoring conditions

AbstractIn this article, we use the Nehari manifold and the eigenvalue problem for the negative Laplacian with Dirichlet boundary condition to analytically study the minimizers for the de Gennes–Cahn–Hilliard energy with quartic double‐well potential and Dirichlet boundary condition on the bounded domain. Our analysis reveals a bifurcation phenomenon determined by the boundary value and a bifurcation parameter that describes the thickness of the transition layer that segregates the binary mixture's two phases. Specifically, when the boundary value aligns precisely with the average of the pure phases, and the bifurcation parameter surpasses or equals a critical threshold, the minimizer assumes a unique form, representing the homogeneous state. Conversely, when the bifurcation parameter falls below this critical value, two symmetric minimizers emerge. Should the boundary value be larger or smaller from the average of the pure phases, symmetry breaks, resulting in a unique minimizer. Furthermore, we derive bounds of these minimizers, incorporating boundary conditions and features of the de Gennes–Cahn–Hilliard energy.

Mathematical Simulation for MHD Casson Convective Nanofluid Flow Induced by 3D Permeable Sheet with Chemical Effect

Objectives: Current manuscript focuses on examination of chemical reaction and heat generation impacts on 3D MHD non-Newtonian nanofluid flow with convective boundary conditions induced by permeable sheet. Additionally, Brownian motion, non-Newtonian heating and thermophoretic processes as used for this study. Methods: A computational programme, MATLAB has been used for solving the system of O.D.Es with the help of ODE45 solver. The Runge Kutta Fehlberg approach is implemented to calculate the answer to the expression for temperature, velocity, and nanoparticle concentration after the shooting process. Findings: For a variety of fluid parameters, the temperature, concentration of nanoparticles, and dimensionless velocities are shown and examined, including permeability parameter , magnetic , stretching ratio parameter , Lewis number , Brownian motion and Prandtl number , thermal Biot number , Casson fluid parameter , chemical reaction parameter . The temperature is found to increase with an enhance in the thermal Biot number and to reduce with a greater Prandtl number and stretching ratio parameter. Novelty: Although the immense significance and frequent use of nanofluids in industries and technology, no effort has been made to explore the chemical influence on MHD Casson fluid flow using a three-dimensional permeable sheet. Through similarity transformations, the Runge-Kutta Fehlberg technique converts mass, momentum, and energy conservation equations into ODEs and incorporates boundary conditions. Skin friction and the heat transmission rate past an extending surface, which have an impact on technology and production, can be predicted using the results of this study. Keywords: Chemical reaction, Buongiorno's model, Nanofluid, Biot numbers, 3D permeable sheet

Simulation Analysis of the Influence of Amplitude on Deformation and Fracture Characteristics of Hard Rock under Ultrasonic Vibration Load

The utilization of auxiliary tools employing ultrasonic high-frequency vibration to enhance rock breaking efficiency holds significant potential for application in underground hard rock excavation engineering. To investigate the failure mechanism of rocks under high frequency ultrasonic vibration load, this study employs particle flow software PFC2D for numerical simulation. By incorporating boundary conditions from actual ultrasonic vibration rock breaking experiments and utilizing a parallel bond model to construct the rock, we analyze the deformation, damage, fracture, and energy evolution process of hard rocks subjected to vibrational loads. The results demonstrate that the maximum displacement in hard rocks increases nearly linearly with vibrations until reaching 5.0199 × 10−3 m, after which it plateaus. Additionally, macroscopic fissures formed during rock failure exhibit an X-shaped pattern. Furthermore, based on our model, we examine the impact of amplitude variation on hard rocks with an equal number of cycles (5,000,000 cycles). Under ultrasonic vibration loads, amplitude influences the total input energy within the rock system. While increasing amplitude does not alter maximum deformation in rocks, it enhances fragmentation degree, fracture degree and energy dissipation coefficient—thereby improving rock breaking efficiency.

Stability and instability of strange dwarfs

Aims. More than 20 years ago, the existence of stable white dwarfs with a core of strange quark matter was proposed. More recently, via the study of radial modes, it has been concluded instead that such objects are unstable. We aim to clarify this issue. Methods. We investigated the stability of these objects by looking at their radial oscillations while incorporating boundary conditions at the quark–hadron interface, which correspond to either a rapid or a slow conversion of hadrons into quarks. Results. Our analysis shows that objects of this type are stable if the star is not strongly perturbed and ordinary matter cannot transform into strange quark matter because of the Coulomb barrier separating the two components. On the other hand, ordinary matter can be transformed into strange quark matter if the star undergoes a violent process, as in the preliminary stages of a type Ia supernova, and this causes the system to become unstable and collapse into a strange quark star. In this way, the accretion-induced collapse of strange dwarfs can be facilitated, and kilometre-sized objects with sub-solar masses can be produced.

Adaptive penalty method with adam optimizer for enhanced convergence in optical waveguide mode solvers.

We propose a cutting-edge penalty method for optical waveguide mode solvers, integrating the Adam optimizer into pseudospectral frequency-domain (PSFD) frameworks. This strategy enables adaptable boundary fluctuations at material interfaces, significantly enhancing numerical convergence and stability. The Adam optimizer, an adaptive algorithm, is deployed to determine the penalty coefficient, greatly improving convergence rates and robustness while effectively incorporating boundary conditions into the interfaces of subdomains. Our solver evaluates the numerical performance of optical waveguides by calculating effective indices of standard benchmark waveguides with high accuracy. This method diminishes numerical boundary errors and provides a marked increase in convergence speed and superior accuracy when compared to conventional methods and even metaheuristic optimization methods, all while maintaining the inherent global spectral accuracy of the PSFD.

Multiscale hydrodynamics modeling reveals the temperature moderating role of the Northern Red Sea Islands

A growing interest in the hydrodynamics of the Red Sea has been observed since the beginning of the 21st century. However, the interaction between the Gulf of Suez (GOS) and the Red Sea along with possible natural mitigation mechanisms of heat stress on its southern coral reef zones have not been adequately investigated. This study evaluated different Regional Ocean Modeling System (ROMS) simulations of the Red Sea using a nesting approach in the southern parts of the GOS to elucidate the three-dimensional nature of thermal variability. The developed regional ROMS model simulated the general circulation patterns and sea surface temperature on the TSUBAME 3.0 supercomputer operated by the Tokyo Institute of Technology. Ultimately, remotely sensed satellite data of Sea Surface Temperature (SST) spanning the period 2016–2020 were used to validate the regional model results. A further challenge posed by the scarcity of distributed depth-varying temperature data on the northern islands' region was overcome by using an offline nesting approach (i.e., incorporating boundary conditions from the parent domain) to simulate the local 3-D thermal regimes. Intriguingly, the results of the nested model scenarios confirmed unique northern islands-enhanced thermal moderating mechanisms where islands act as barriers to the impacts of the relatively warmer water originating from the eastern boundary current. Additionally, this study introduces a new approach to applying higher-resolution models to the precise spatial and temporal representation of thermal indices in a way that surpasses the widely adopted remote sensing approaches. In short, multiscale modeling provides a valuable approach for assessing the thermal regimes around one of the most precious marine ecosystems in the world.

Heat transfer of MHD flow over a wedge with a surface of mutable temperature

AbstractThe objective of this study is to investigate the thermal distribution and heat transfer in the boundary layer of a wedge with a variable surface temperature in the presence of a magnetic field. To achieve this, we first used similarity solutions to transform the governing equations of magnetohydrodynamic flow for variable surface temperature conditions into ordinary differential equations. We then solved the resulting equations using the collocation method (CM) with different intensity magnetic fields and varying Hartmann (Ha) numbers and surface temperatures. The CM was further modified by incorporating boundary conditions. The results obtained from the solved equations were validated and compared with those obtained using the numerical Runge–Kutta fourth‐order method and previous literature. Finally, we investigated the impact of various parameters on the friction coefficient (Cf) and Nusselt number (Nu), including the power of variable surface temperature (n), Prandtl (Pr) number, Eckert (Ec) number, the half angle of the wedge (φ), and Ha number. We considered values for these parameters within the ranges 0.5 ≤ n ≤ 1.5, 0.5 ≤ Pr ≤ 5, 0.001 ≤ Ec ≤ 0.002, 15° ≤ φ ≤ 60°, and 0 ≤ Ha ≤ 3. Our findings indicate that the slope of the boundary layer increases with increasing Ha or φ, resulting in an increase in Cf on the surface by up to 526%. The Nu number, calculated using the energy equation, increases up to 91.7%, 39.8%, and 1.43% with increasing Ha, n, and Ec, respectively, resulting in faster growth of the thermal boundary layer, which causes the thickness to decrease and the Nu number to rise. However, as φ increases, the Nu number drops on the surface, and the heat transfer behavior remains similar to that observed previously.

Computational fluid dynamics model to predict the dynamical behavior of the cerebrospinal fluid through implementation of physiological boundary conditions.

Cerebrospinal fluid (CSF) dynamics play an important role in maintaining a stable central nervous system environment and are influenced by different physiological processes. Multiple studies have investigated these processes but the impact of each of them on CSF flow is not well understood. A deeper insight into the CSF dynamics and the processes impacting them is crucial to better understand neurological disorders such as hydrocephalus, Chiari malformation, and intracranial hypertension. This study presents a 3D computational fluid dynamics (CFD) model which incorporates physiological processes as boundary conditions. CSF production and pulsatile arterial and venous volume changes are implemented as inlet boundary conditions. At the outlets, 2-element windkessel models are imposed to simulate CSF compliance and absorption. The total compliance is first tuned using a 0D model to obtain physiological pressure pulsations. Then, simulation results are compared with in vivo flow measurements in the spinal subarachnoid space (SAS) and cerebral aqueduct, and intracranial pressure values reported in the literature. Finally, the impact of the distribution of and total compliance on CSF pressures and velocities is evaluated. Without respiration effects, compliance of 0.17ml/mmHg yielded pressure pulsations with an amplitude of 5mmHg and an average value within the physiological range of 7-15mmHg. Also, model flow rates were found to be in good agreement with reported values. However, when adding respiration effects, similar pressure amplitudes required an increase of compliance value to 0.51ml/mmHg, which is within the range of 0.4-1.2ml/mmHg measured in vivo. Moreover, altering the distribution of compliance over the four different outlets impacted the local flow, including the flow through the foramen magnum. The contribution of compliance to each outlet was directly proportional to the outflow at that outlet. Meanwhile, the value of total compliance impacted intracranial pressure. In conclusion, a computational model of the CSF has been developed that can simulate CSF pressures and velocities by incorporating boundary conditions based on physiological processes. By tuning these boundary conditions, we were able to obtain CSF pressures and flows within the physiological range.

Reliability of using generic flow conditions to quantify aneurysmal haemodynamics: A comparison against simulations incorporating boundary conditions measured in vivo

Background and objectivesInitiation, growth, and rupture of intracranial aneurysms are believed to be closely related to their local haemodynamic environment. While haemodynamics can be characterised by use of computational fluid dynamics (CFD), its reliability depends heavily upon accurate assumption of the boundary conditions. Herein, we compared the simulated aneurysmal haemodynamics obtained by use of generic boundary conditions against those obtained under flow conditions measured in vivo. MethodsWe prospectively recruited 19 patients with intracranial aneurysms requiring 3-dimensional rotational angiography, during which blood pressure at the internal carotid artery was probed by catheter and flowrate measured by a dedicated software tool. Using these flow conditions measured in vivo, we quantified the aneurysmal haemodynamics for each patient by CFD, and then compared the results with those derived from a generic condition reported in the literature, in terms of the time-averaged wall shear stress (TAWSS), oscillatory shear index (OSI), relative residence time (RRT), and percentage of the intra-aneurysmal flow (PIAF). In addition, the effects on aneurysmal haemodynamics of different outflow strategies (splitting method vs. Murray's Law) and simulation schemes (transient vs. steady-state) relative to each flow condition were also assessed. ResultsDifferences in the simulated TAWSS (−6.08 ± 10.64 Pa, p = 0.001), OSI (0.06 ± 0.13, p = 0.001), and PIAF (−0.05 ± 0.20, p = 0.012) between the patient-specific and generic boundary conditions were found to be statistically significant, in contrast to that in the RRT (49 ± 307 Pa−1, p = 0.062). Outflow strategies did not yield statistically significant differences in any of the investigated parameters (all p > 0.05); rather, the resulting parameters were found to be in good correlations (all r > 0.71, p < 0.001). Difference between the aneurysmal TAWSS and the WSS derived from cycle-averaged flowrate condition was found to be minor (0.66 ± 1.36 Pa, p = 0.000), so was that between PIAFs obtained respectively from the transient and steady-state simulations (0.02 ± 0.05, p = 0.000). ConclusionsIncorporating into simulation the patient-specific boundary conditions is critical for CFD to characterise aneurysmal haemodynamics, while outflow strategies may not introduce significant uncertainties. Steady-state simulation incorporating the cycle-averaged flow condition may produce unbiased WSS and PIAF compared to the transient analysis.

A Novel Approach to Incorporate Boundary Conditions in Rectangular and Radial-Cylindrical Coordinates for Block Centered-Grid in Reservoir Simulation

A Novel Approach to Incorporate Boundary Conditions in Rectangular and Radial-Cylindrical Coordinates for Block Centered-Grid in Reservoir Simulation

Bayesian-entropy gaussian process for constrained metamodeling

A novel Bayesian-Entropy Gaussian Process (BEGP) is proposed for constrained metamodeling. Gaussian Process (GP) regression is a flexible and robust tool for surrogate modeling using observation data. For many engineering problems, available information other than observations may be known, such as physical constraints, boundary conditions, and empirical knowledge. Based on the Bayesian-Entropy (BE) principle, this paper introduces a novel framework for encoding extra information in addition to point data in constructing a GP regression model. The extra information is treated as constraints on the mean prediction of GP. The BE method can rigorously incorporate extra information as constraints into classical Bayesian framework. The constraint term is added into the posterior distribution of the hyperparameters when training the GP model. BEGP serves as an information fusion tool to enhance the extrapolation behavior of the GP model by incorporating additional knowledge about the problem. The proposed methodology is demonstrated on a numerical toy example and a structural analysis example highlighting the ability to smoothly connect two local GPs and incorporate boundary conditions as extra constraints. The BEGP shows the ability of incorporating physics constraints to enhance prediction and extrapolation behaviors. Finally, conclusions and future work are drawn based on the proposed study.

Complex Rayleigh Waves in Nonhomogeneous Magneto-Electro-Elastic Half-Spaces.

The complex Rayleigh waves play an important role in the energy conversion efficiency of magneto-electro-elastic devices, so it is necessary to explore the wave propagation characteristics for the better applications in engineering. This paper modifies the Laguerre orthogonal polynomial to investigate the complex Rayleigh waves propagating in nonhomogeneous magneto-electro-elastic half-spaces. The improved method simplifies the calculation process by incorporating boundary conditions into the constitutive relations, shortens the solving time by transforming the solution of wave equation to an eigenvalue problem, and obtains all wave modes, including real and imaginary and complex wavenumbers. The three-dimensional curves of full frequency spectrum and phase velocities are presented for the better description of the conversion from complex Rayleigh wave modes to real wave ones; besides, the displacement distributions, electric and magnetic potential curves are obtained in thickness and propagation directions, respectively. Numerical results are analyzed and discussed elaborately in three cases: variation of nonhomogeneous coefficients, absence of magnetism, and absence of electricity. The results can be used to optimize and fabricate the acoustic surface wave devices of the nonhomogeneous magneto-electro-elastic materials.

Anisotropic multiscale systems on bounded domains

We establish a construction of hybrid multiscale systems on a bounded domain ${\Omega } \subset \mathbb {R}^{2}$ consisting of shearlets and boundary-adapted wavelets, which satisfy several properties advantageous for applications to imaging science and the numerical analysis of partial differential equations. More precisely, we construct hybrid shearlet-wavelet systems that form frames for the Sobolev spaces $H^{s}({\Omega }),~s\in \mathbb {N} \cup \{0\}$ with controllable frame bounds and admit optimally sparse approximations for functions which are smooth apart from a curve-like discontinuity. Per construction, these systems allow to incorporate boundary conditions.

Error estimation in reduced basis method for systems with time-varying and nonlinear boundary conditions

Many physical phenomena, such as mass transport and heat transfer, are modeled by systems of partial differential equations with time-varying and nonlinear boundary conditions. Control inputs and disturbances typically affect the system dynamics at the boundaries and a correct numerical implementation of boundary conditions is therefore crucial. However, numerical simulations of high-order discretized partial differential equations are often too computationally expensive for real-time and many-query analysis. For this reason, model complexity reduction is essential. In this paper, it is shown that the classical reduced basis method is unable to incorporate time-varying and nonlinear boundary conditions. To address this issue, it is shown that, by using a modified surrogate formulation of the reduced basis ansatz combined with a feedback interconnection and a input-related term, the effects of the boundary conditions are accurately described in the reduced-order model. The results are compared with the classical reduced basis method. Unlike the classical method, the modified ansatz incorporates boundary conditions without generating unphysical results at the boundaries. Moreover, a new approximation of the bound and a new estimate for the error induced by model reduction are introduced. The effectiveness of the error measures is studied through simulation case studies and a comparison with existing error bounds and estimates is provided. The proposed approximate error bound gives a finite bound of the actual error, unlike existing error bounds that grow exponentially over time. Finally, the proposed error estimate is more accurate than existing error estimates.

Incorporating boundary conditions in a stochastic volatility model for the numerical approximation of bond prices

In this paper, we consider a two‐factor interest rate model with stochastic volatility, and we assume that the instantaneous interest rate follows a jump‐diffusion process. In this kind of problems, a two‐dimensional partial integro‐differential equation is derived for the values of zero‐coupon bonds. To apply standard numerical methods to this equation, it is customary to consider a bounded domain and incorporate suitable boundary conditions. However, for these two‐dimensional interest rate models, there are not well‐known boundary conditions, in general. Here, in order to approximate bond prices, we propose new boundary conditions, which maintain the discount function property of the zero‐coupon bond price. Then, we illustrate the numerical approximation of the corresponding boundary value problem by means of an alternative direction implicit method, which has been already applied for pricing options. We test these boundary conditions with several interest rate pricing models.

Cubic spline wavelets with four vanishing moments on the interval and their applications to option pricing under Kou model

The paper is concerned with the construction of a cubic spline wavelet basis on the unit interval and an adaptation of this basis to the first-order homogeneous Dirichlet boundary conditions. The wavelets have four vanishing moments and they have the shortest possible support among all cubic spline wavelets with four vanishing moments corresponding to B-spline scaling functions. We provide a rigorous proof of the stability of the basis in the space [Formula: see text] or its subspace incorporating boundary conditions. To illustrate the applicability of the constructed bases, we apply the wavelet-Galerkin method to option pricing under the double exponential jump-diffusion model and we compare the results with other cubic spline wavelet bases and with other methods.

Doubling the Convergence Rate by Pre- and Post-Processing the Finite Element Approximation for Linear Wave Problems

In this paper, a novel pre- and post-processing algorithm is presented that can double the convergence rate of finite element approximations for linear wave problems. In particular, it is shown that a $q$-step pre- and post-processing algorithm can improve the convergence rate of the finite element approximation from order $p+1$ to order $p+1+q$ in the $L^2$-norm and from order $p$ to order $p+q$ in the energy norm, in both cases up to a maximum of order $2p$, with $p$ the polynomial degree of the finite element space. The $q$-step pre- and post-processing algorithms only need to be applied once and require solving at most $q$ linear systems of equations. The biggest advantage of the proposed method compared to other post-processing methods is that it does not suffer from convergence rate loss when using unstructured meshes. Other advantages are that this new pre- and post-processing method is straightforward to implement, incorporates boundary conditions naturally, and does not lose accuracy near boundaries or strong inhomogeneities in the domain. Numerical examples illustrate the improved accuracy and higher convergence rates when using this method. In particular, they confirm that $2p$-order convergence rates in the energy norm are obtained, even when using unstructured meshes or when solving problems involving heterogeneous domains and curved boundaries.

Incorporating reflection boundary conditions in the Neumann series radiative transport equation: application to photon propagation and reconstruction in diffuse optical imaging.

We propose a formalism to incorporate boundary conditions in a Neumann-series-based radiative transport equation. The formalism accurately models the reflection of photons at the tissue-external medium interface using Fresnel's equations. The formalism was used to develop a gradient descent-based image reconstruction technique. The proposed methods were implemented for 3D diffuse optical imaging. In computational studies, it was observed that the average root-mean-square error (RMSE) for the output images and the estimated absorption coefficients reduced by 38% and 84%, respectively, when the reflection boundary conditions were incorporated. These results demonstrate the importance of incorporating boundary conditions that model the reflection of photons at the tissue-external medium interface.

Geothermal heat flux in the Amundsen Sea sector of West Antarctica: New insights from temperature measurements, depth to the bottom of the magnetic source estimation, and thermal modeling

AbstractFocused research on the Pine Island and Thwaites glaciers, which drain the West Antarctic Ice Shelf (WAIS) into the Amundsen Sea Embayment (ASE), revealed strong signs of instability in recent decades that result from variety of reasons, such as inflow of warmer ocean currents and reverse bedrock topography, and has been established as the Marine Ice Sheet Instability hypothesis. Geothermal heat flux (GHF) is a poorly constrained parameter in Antarctica and suspected to affect basal conditions of ice sheets, i.e., basal melting and subglacial hydrology. Thermomechanical models demonstrate the influential boundary condition of geothermal heat flux for (paleo) ice sheet stability. Due to a complex tectonic and magmatic history of West Antarctica, the region is suspected to exhibit strong heterogeneous geothermal heat flux variations. We present an approach to investigate ranges of realistic heat fluxes in the ASE by different methods, discuss direct observations, and 3‐D numerical models that incorporate boundary conditions derived from various geophysical studies, including our new Depth to the Bottom of the Magnetic Source (DBMS) estimates. Our in situ temperature measurements at 26 sites in the ASE more than triples the number of direct GHF observations in West Antarctica. We demonstrate by our numerical 3‐D models that GHF spatially varies from 68 up to 110 mW m−2.