Stationary Equation Research Articles

Variational–hemivariational system for contaminant convection–reaction–diffusion model of recovered fracturing fluid

Abstract This work is devoted to study the convection–reaction–diffusion behavior of contaminant in the recovered fracturing fluid which flows in the wellbore from shale gas reservoir. First, we apply various constitutive laws for generalized non-Newtonian fluids, diffusion principles, and friction relations to formulate the recovered fracturing fluid model. The latter is a partial differential system composed of a nonlinear and nonsmooth stationary incompressible Navier-Stokes equation with a multivalued friction boundary condition, and a nonlinear convection–reaction–diffusion equation with mixed Neumann boundary conditions. Then, we provide the weak formulation of the fluid model which is a hemivariational inequality driven by a nonlinear variational equation. We establish existence of solutions to the recovered fracturing fluid model via a surjectivity theorem for multivalued operators combined with an alternative iterative method and elements of nonsmooth analysis.

Extreme dry advection dominates the record-breaking Yangtze River heatwave in midsummer of 2022

The Yangtze River Valley (YRV) experienced an unprecedented heatwave in midsummer of 2022. Still, the detailed physical processes involved in the influence of abnormal large-scale atmospheric circulation on the heatwave remain unexplored. Here, we show that the positive meridional gradient of anomalous atmospheric moisture at the middle-lower troposphere and associated extreme dry air advection over the YRV are vital prerequisites for forming the 2022 YRV heatwave. The 2022 YRV heatwave is dominated by interannual variability, contributing 72.7% to the total temperature anomalies. Diagnosis of the surface heat budget equation indicates that the surface cloud radiative forcing is the most critical process in driving the 2022 YRV heatwave, which is dominated by the positive surface short-wave cloud radiative forcing associated with the suppressed precipitation and the middle-low clouds. The suppressed precipitation is induced by the vertical dynamical processes of anomalous moisture advection caused by the abnormal descending flows over the YRV, which are driven by the negative advection of anomalous latent heat energy by climatological meridional wind (anomalous dry air advection) according to the atmospheric moist static energy equation. Simulations from the Lagrangian model FLEXPART further indicate that the moisture anomaly over the north of YRV mainly originated from the surface evaporation in the YRV, implying that there is a positive land-air feedback during the life cycle of the YRV heatwave. Our study enriches the mechanism understanding of the 2022 YRV heatwave from the perspective of surface energy budget and land-air feedback.

Analysis of a two phase flow model of biofilm spread

We consider a quasi-stationary problem describing the status of velocities, pressures and chemicals affecting cell behavior within a biofilm. The model couples stationary transport equations and compressible Stokes systems with convection–reaction–diffusion equations. We establish existence, uniqueness and stability of solutions of the different submodels involved and then obtain well posedness results for the quasi-stationary system. Our analysis relies on the construction of weak solutions for the steady transport equations under sign assumptions and the reformulation of the compressible Stokes problem as an elliptic system with enhanced regularity properties on the pressure. We need to consider velocity fields whose divergence and normal boundary components satisfy sign conditions. Applications include the study of cells, biofilms and tissues, where one phase is a liquid solution, whereas the other one is assorted biomass.

A Banach spaces-based fully mixed virtual element method for the stationary two-dimensional Boussinesq equations

In this paper we extend recent results obtained for the Navier–Stokes equations to propose and analyze a new fully mixed virtual element method (mixed-VEM) for the stationary two-dimensional Boussinesq equations appearing in non-isothermal flow phenomena. The model consists of a Navier–Stokes type system, modeling the velocity and the pressure of the fluid, coupled to an advection-diffusion equation for the temperature. The variational formulation is based on the introduction of the additional unknowns given by a modified pseudostress tensor, which depends on the pressure, and the diffusive and convective terms of the fluid, and the pseudoheat vector, which involves the temperature, its gradient, and the velocity. As a consequence of the former, the pressure is eliminated from the system, but computed afterwards via a post-processing formula. In turn, for the Galerkin approximation we follow the approach employed in a previous work introducing for the first time an Lp spaces-based mixed-VEM for the Navier–Stokes equations, and couple it with a similar mixed-VEM for the convection–diffusion equation modeling the temperature. The solvability analysis of the resulting coupled discrete scheme is carried out by using appropriate fixed-point arguments, along with the discrete versions of the Babuška–Brezzi theory and the Banach-Nečas-Babuška theorem, both in subspaces of Banach spaces. The first Strang lemma is applied to derive the a priori error estimates for the virtual element solution as well as for the fully computable approximation of the pseudostress tensor, the pseudoheat vector, and the post-processed pressure. Finally, several numerical results, illustrating the performance of the mixed-VEM scheme and confirming the rates of convergence predicted by the theory, are reported.

Symmetry Analysis of the Two-Dimensional Stationary Gas Dynamics Equations in Lagrangian Coordinates

This article analyzes the symmetry of two-dimensional stationary gas dynamics equations in Lagrangian coordinates, including the search for equivalence transformations, the group classification of equations, the derivation of group foliations, and the construction of conservation laws. The consideration of equations in Lagrangian coordinates significantly simplifies the procedure for obtaining conservation laws, which are derived using the Noether theorem. The final part of the work is devoted to group foliations of the gas dynamics equations, including for the nonstationary isentropic case. The group foliations approach is usually employed for equations that admit infinite-dimensional groups of transformations (which is exactly the case for the gas dynamics equations in Lagrangian coordinates) and may make it possible to simplify their further analysis. The results obtained in this regard generalize previously known results for the two-dimensional shallow water equations in Lagrangian coordinates.

Statics and dynamics of pulley-driven tensegrity structures with sliding cable modeling

This article introduces the concept of Pulley-Driven Clustered Tensegrity Structures (PD-CTS) and develops nonlinear and linearized static and dynamic equations using the Lagrangian method. The generalized coordinates utilized in this framework comprise the nodal coordinates of the tensegrity structure and the string sliding distances. The governing equations are specifically constructed, considering situations with and without boundary constraints. The developed method can be applied to nonlinear and linearized static and dynamic analysis of any PD-CTS, covering various research domains such as load analysis, form-finding, large deformation analysis, modal analysis, and dynamic response to various inputs. The substructure method is used to lower the order of the equations to reduce computational costs and facilitate the study of actuation strategies of tensegrity structures. The effectiveness of the suggested methods is validated through the analysis of two case studies: a 2D T-bar and a 3D tensegrity tower. Furthermore, the methodologies established in this paper can also be employed to examine deployable cable net structures and pulley-driven robotic systems.

Research on numerical solution of statically indeterminate problem of steering tool combined bearing

The steering tool combined bearing is a mechanical component that can support forces, guide movement, and bear vibrations at the same time. In the process of analyzing the contact and load distribution inside the combined bearing, there are several contact points, and the static equilibrium equation is statically indeterminate. Based on dynamics theory and Hertz elastic contact theory, the concept of a load integral and the displacement estimation formula for roller bearings are introduced, and a numerical solution to the statically indeterminate problem of the steering tool combined bearing is given. According to the statically indeterminate equation matrix of the combined bearing, the relationship between the load and displacement of the combined bearing is studied to improve the statically indeterminate equation matrix. Moreover, the simulation results of the maximum stress and displacement distribution results of the inner and outer ring eccentricity from 1 to 6mm are compared with the algorithm prediction results. The error between the numerical solution of the statically indeterminate problem of the steering tool combined bearing and the simulation result is small, improving the calculation efficiency. This provides theoretical support for the design and reasonable application of the steering tool combined bearing.

Efficient stochastic modal decomposition methods for structural stochastic static and dynamic analyses

AbstractThis article presents unified and efficient stochastic modal decomposition methods to solve stochastic structural static and dynamic problems. We extend the idea of deterministic modal decomposition method for structural dynamic analysis to stochastic cases. Standard/generalized stochastic eigenvalue equations are adopted to calculate the stochastic subspaces for stochastic static/dynamic problems and they are solved by an efficient reduced‐order method. The stochastic solutions of both static and dynamic equations are approximated by stochastic bases of the stochastic subspaces. Original stochastic static/dynamic equations are then transformed into a set of single‐degree‐of‐freedom (SDoF) stochastic static/dynamic equations, which are efficiently solved by the proposed non‐intrusive methods. Specifically, a non‐intrusive stochastic Newmark method is developed for the solution of SDoF stochastic dynamic equations, and the element‐wise division of sample vectors is used to solve the SDoF stochastic static equations. All of these methods have low computational effort and are weakly sensitive to the stochastic dimension, thus the proposed methods avoid the curse of dimensionality successfully. Two numerical examples, including two‐ and three‐dimensional spatial problems with low and high stochastic dimensions, are given to show the efficiency and accuracy of the proposed methods.

Thermoelastic analysis of axisymmetric conical shells: Investigating stress–strain response under uniform heat flow with semi-coupled approach

ABSTRACT This study addresses the issue of thermal stresses and strains in a thin-walled conical shell subjected to uniform heat flow along its side surfaces and at both ends of the thermal insulation shell. The governing equations are derived using a semi-coupled static thermoelastic equation and an energy equation. The energy equation ignores strain rate effects, but static behavior equations consider temperature variations in axisymmetric thin-walled conical shells. Semicoupled methods, due to their simpler analysis, are suitable for thermal stress analysis when structural and thermal disturbances differ significantly. To obtain solutions, the Galerkin finite element method is employed, yielding valuable insights. Using the Galerkin reduced weight residual method, the sum of the weight residuals on the shell thickness is set to zero. Through a comprehensive problem-solving approach, the effects of increasing the angle of the cone apex in different thermal boundary conditions are investigated in two one-ended and two-ended cases.

Variational Principles in Quaternionic Analysis with Applications to the Stationary MHD Equations

In this paper we aim to combine tools from variational calculus with modern techniques from quaternionic analysis that involve Dirac type operators and related hypercomplex integral operators. The aim is to develop new methods for showing geometry independent explicit global existence and uniqueness criteria as well as new computational methods with special focus to the stationary incompressible viscous magnetohydrodynamic equations. We first show how to specifically apply variational calculus in the quaternionic setting. To this end we explain how the mountain pass theorem can be successfully applied to guarantee the existence of (weak) solutions. To achieve this, the quaternionic integral operator calculus serves as a key ingredient allowing us to apply Schauder’s fixed point theorem. The advantage of the approach using Schauder’s fixed point theorem is that it is also applicable to large data since it does not require any kind of contraction property. These considerations will allow us to provide explicit iterative algorithms for its numerical solution. Finally to obtain more precise a-priori estimates one can use in the situations dealing with small data the Banach fixed point theorem which then also grants the uniqueness.

Numerical Study on the Precipitation Concentration over the Western Coast of Sumatra Island

Abstract Large amounts of tropical precipitation have been observed as significantly concentrated over the western coast of Sumatra Island. In the present study, we used a cloud-resolving model to perform 14-day numerical simulations and reproduce the distinctive precipitation distributions over western Sumatra Island and adjacent areas. The control experiment, in which the warmer sea surface temperature (SST) near the coast was incorporated and the terminal velocity and effective radius of ice clouds were parameterized to be temperature dependent, adequately reproduced the precipitation concentration as well as the diurnal cycles of precipitation. We then used the column-integrated frozen moist static energy budget equation, which is virtually equivalent to the column-integrated moisture budget equation under the weak temperature gradient assumption, to formulate sensitivity experiments focusing on the effects of coastal SST and upper-level ice clouds. Analysis of the time-averaged fields revealed that the column-integrated moisture and precipitation in the coast were significantly reduced when a cooler coastal SST or larger ice cloud particle size was assumed. Based on the comparison of the sensitivity experiments and in situ observations, we speculate that ice clouds, which are exported from inland convection that is strictly regulated by solar radiation, promote the accumulation of moisture in the coastal region by mitigating radiative cooling. Together with the moisture and heat supplied by the warm ocean surface, they contribute to the large amounts of precipitation here.

Anomalous Magnetization Spikes in the Locally Corrected Nyström Discretization of Static Volume Integral Equation using Tetrahedral Cells

A locally corrected Nyström (LCN) discretization of a magnetostatic volume integral equation is presented. Anomalous magnetization spikes can occur when the underlying mesh uses tetrahedral cells regardless of discretization order. The mechanism for the anomalous magnetization spikes is discussed, and mitigation of the spikes through use of an LCN-to-Moment Method conversion is investigated. Results are presented validating that the LCN-to-Moment Method suppresses the anomalous spikes.

Projections of Drug-Drug Interactions Caused by Time-Dependent Inhibitors of Cytochrome P450 1A2, 2B6, 2C8, 2C9, 2C19, and 2D6 Using In Vitro Data in Static and Dynamic Models.

In vitro time-dependent inhibition (TDI) kinetic parameters for cytochrome P450 (CYP) 1A2, 2B6, 2C8, 2C9, 2C19, and 2D6, were determined in pooled human liver microsomes for 19 drugs (and 2 metabolites) for which clinical drug-drug interactions (DDI) are known. In vitro TDI data were incorporated into the projection of the magnitude of DDIs using mechanistic static models and Simcyp®. Results suggest that for the mechanistic static model, use of estimated average unbound exit concentration of the inhibitor from the liver resulted in a successful prediction of observed magnitude of clinical DDIs and was similar to Simcyp®. Overall, predictions of DDI magnitude (i.e., fold increase in AUC of a CYP-specific marker substrate) were within 2-fold of actual values. Geometric mean-fold errors were 1.7 and 1.6 for static and dynamic models, respectively. Projections of DDI from both models were also highly correlated to each other (r2 = 0.92). This investigation demonstrates that DDI can be reliably predicted from in vitro TDI data generated in HLM for several CYP enzymes. Simple mechanistic static model equations as well as more complex dynamic PBPK models can be employed in this process. Significance Statement Cytochrome P450 time-dependent inhibitors (TDI) can cause drug-drug interactions (DDI). An ability to reliably assess the potential for a new drug candidate to cause DDI is essential during drug development. In this report, TDI data for 19 drugs (and 2 metabolites) were measured and used in static and dynamic models to reliably project the magnitude of DDI resulting from inhibition of CYP1A2, 2B6, 2C8, 2C9, 2C19, and 2D6.

Suspension and tyre loads estimation of an FSAE car: model development and on-track validation

In this paper a relatively simple model is presented, able to compute the loads on the suspension arms and at the tyres' contact patches of a Formula SAE car starting from telemetry data. The model is based on standard dynamic equilibrium equations, a simplified assumption for the limited-slip differential and a given front-to-rear brake distribution. The model inputs are the signals acquired with common telemetry sensors, which usually equip a Formula SAE vehicle (Inertia Measurement Unit, GPS, potentiometers, Hall sensors). In order to validate the model, some arms of the suspensions were instrumented with strain gauges, and a kinematic model of the suspension, together with its static equilibrium equation set, was set up to relate the arm loads to the forces acting at the tyre-road contact patch. The suspension kinematic and equilibrium models were first validated through static tests, with known forces applied at the wheel hub. Then, the complete vehicle model was validated by comparing the same quantities, relative to some real driving tests on a kart circuit, showing a fairly good agreement between the predicted and the measured loads. The model represents a valuable tool for vehicle design and development for the Formula SAE team.

Negative flows and non-autonomous reductions of the Volterra lattice

We study reductions of the Volterra lattice corresponding to stationary equations for the additional, noncommutative subalgebra of symmetries. It is shown that, in the case of general position, such a reduction is equivalent to the stationary equation for a sum of the scaling symmetry and the negative flows, and is written as $(m+1)$-component difference equations of the Painlev\'e type generalizing the dP$_1$ and dP$_{34}$ equations. For these reductions, we present the isomonodromic Lax pairs and derive the B\"acklund transformations which form the $\mathbb{Z}^m$ lattice.

Some remarks about the stationary micropolar fluid equations: Existence, regularity and uniqueness

We consider here the stationary micropolar fluid equations, which are a special generalization of the usual Navier-Stokes system where the microrotations of the fluid particles must be taken into account. We thus obtain two coupled equations: one mainly based on the velocity field u→ and the other on the microrotation field ω→. In this work we will study some problems related to the existence of weak solutions as well as some regularity and uniqueness properties. Our main result establishes the uniqueness of the trivial solution under some suitable infinity decay conditions for the velocity field only.

IPMC-based actuators: An approach for measuring a linear form of its static equation

Ionic Polymer-Metal Composites (IPMCs) are smart materials used as actuators, sensors, and energy harvesters. They are known as a subset of Electroactive Polymers (EAPs). IPMCs structure is layered with one polymer layer and two of electrodes. In this paper, the actuation behavior of an IPMC sample with platinum electrodes and Nafion-117 polymer is of the interest and two diverse methods have been applied; This microelectromechanical system which is used in this paper is manufactured initially, and then the experimental method was applied besides the finite element method. According to the experimental analysis, the maximum displacement of a cantilever model of the IPMC in various lengths and different amounts of voltages is measured and recorded. The experimental method is used to validate the finite element method results. By using Linear Regression with the Ordinary-Least- Squares method, a linear equation is derived that will provide the maximum displacement of a cantilever beam model of the IPMC in varying lengths and different applied Voltages. This equation can be a special-occasion alteration for the specific application of Poisson- Nernst-Plank equation. This multi-input linear equation can predict ionic polymer-metal composite attitude accurately.

Compressive Fourier collocation methods for high-dimensional diffusion equations with periodic boundary conditions

Abstract High-dimensional partial differential equations (PDEs) are a popular mathematical modelling tool, with applications ranging from finance to computational chemistry. However, standard numerical techniques for solving these PDEs are typically affected by the curse of dimensionality. In this work, we tackle this challenge while focusing on stationary diffusion equations defined over a high-dimensional domain with periodic boundary conditions. Inspired by recent progress in sparse function approximation in high dimensions, we propose a new method called compressive Fourier collocation. Combining ideas from compressive sensing and spectral collocation, our method replaces the use of structured collocation grids with Monte Carlo sampling and employs sparse recovery techniques, such as orthogonal matching pursuit and $\ell ^1$ minimization, to approximate the Fourier coefficients of the PDE solution. We conduct a rigorous theoretical analysis showing that the approximation error of the proposed method is comparable with the best $s$-term approximation (with respect to the Fourier basis) to the solution. Using the recently introduced framework of random sampling in bounded Riesz systems, our analysis shows that the compressive Fourier collocation method mitigates the curse of dimensionality with respect to the number of collocation points under sufficient conditions on the regularity of the diffusion coefficient. We also present numerical experiments that illustrate the accuracy and stability of the method for the approximation of sparse and compressible solutions.

New error analysis of charge-conservative finite element methods for stationary inductionless MHD equations

In this paper, we present a new error analysis of a class of charge-conservative finite element methods for stationary inductionless magnetohydrodynamics (MHD) equations. The methods use the standard inf-sup stable Mini/Taylor–Hood pairs to discretize the velocity and pressure, and the Raviart–Thomas face element for solving the current density. Due to the strong coupling of the system and the pollution of the lower-order Raviart–Thomas face approximation in analysis, the existing analysis is not optimal. In terms of a mixed Poisson projection and the corresponding estimate in negative norms, we establish new and optimal error estimates. In particular, we prove that the method with the lowest-order Raviart–Thomas face element and Mini element provides the optimal accuracy for the velocity in L2-norm, and the method with the lowest-order Raviart–Thomas face element and P2−P1 Taylor-Hood element supplies the optimal accuracy for the velocity in H1-norm and the pressure in L2-norm. Furthermore, we propose a simple recovery technique to obtain a new numerical current density of one order higher accuracy by re-solving a mixed Poisson equation. Numerical results are provided to verify the theoretical analysis.

Schauder estimates for stationary and evolution equations associated to stochastic reaction-diffusion equations driven by colored noise

We consider stochastic reaction-diffusion equations with colored noise on the space of real-valued and continuous functions on a compact subset of ℝ d for d = 1 , 2 , 3 . We prove Schauder-type estimates, which will depend on the color of the noise, for the stationary and evolution problems associated with the corresponding transition semigroup.