Stationary Equation Research Articles

Liouville-type theorems for the stationary inhomogeneous incompressible MHD equations

In this paper, we investigate the Liouville-type problem for the three-dimensional stationary inhomogeneous incompressible MHD equations in the Lorentz spaces without any integrability condition for (∇u,∇B). More precisely, we prove that the velocity and magnetic field (u,B), being in some Lorentz spaces, must be zero provided that the density ρ is essentially bounded.

Black lens in a bubble of nothing

Applying the inverse scattering method to static and biaxisymmetric Einstein equations, we construct a nonrotating black lens inside a bubble of nothing whose horizon is topologically lens space, $L(n,1)={S}^{3}/{\mathbb{Z}}_{n}$. Using this solution, we discuss whether a static black lens can be in equilibrium by the force balance between the expansion and gravitational attraction.

Research on Non-Circular Raceway of Single-Row Four-Point Contact Ball Bearing Based on Life Optimization

The traditional slewing bearing with circular raceway has the problem of stress concentration under the large overturning moment. In this paper, a new non-circular raceway of a slewing ball bearing was presented to conquer this problem. To obtain the new non-circular raceway, first, the static equilibrium equations of the slewing ball bearing was established by the vector method, which is a constraint condition for life optimization; secondly, the life optimization function was established to calculate the contact load distribution in the bearing when the bearing life is at its maximum; finally, through the contact deformation with the contact load, the non-circular inner raceway corresponding to the maximum bearing life was obtained, and the non-circular shape corresponding to the raceway under axial load and overturning moment was studied. The results show that the non-circular raceway devised by this method can evenly reduce the contact force of the raceway and effectively improve the bearing capacity. Moreover, the position where the non-circular raceway deformation occurs is the position where the contact force is different before and after optimization. Therefore, the overturning moment has an effect on the shape of the raceway, whereas the axial load only affects the amplitude of the deformation, and has no obvious effect on the shape of the raceway.

A computationally efficient and mechanically compatible multi-phase-field model applied to coherently stressed three-phase solids

Engineering alloys generally exhibit multi-phase microstructures. For simulating their microstructure evolution during solid-state phase transformation, CALPHAD-guided multi-phase-field models coupled with micro-mechanics have proven to be a reliable simulation tool. Nevertheless, their efficiency and accuracy still depend on the homogenization scheme used to interpolate the elastic properties in the interfacial regions. In this paper, we present a phase-field model for multi-phase and multi-component solids using a partial rank-one homogenization scheme that enforces static and kinematic compatibilities in the interfacial regions. To this end, we first extend the rank-one homogenization scheme to multi-phase systems. Moreover, for computational efficiency, we analytically solve the static compatibility equations for linear elastic three-phase solids. For quantitative accuracy, a coupling technique is used to extract the prerequisite thermodynamic and kinetic properties from CALPHAD databases. The model is solved numerically in an open source finite-element framework. As numerical applications, the microstructure of two elastically stressed intermetallic-containing three-phase alloys: Ni–Al and Al–Cr–Ni, are simulated. The accuracy of the model is verified against analytically obtained solutions for planar and concentric ring interfaces. We show that the simulation results remain unaltered with varying interface width. Except for one simulation, all cases show better or nearly equal convergence using the partial rank-one scheme compared to the Voigt–Taylor scheme.

Two-level iterative finite element methods for the stationary natural convection equations with different viscosities based on three corrections

Two-level iterative finite element methods for the stationary natural convection equations with different viscosities based on three corrections

Minimal mass design of clustered tensegrity structures

This paper presents a minimal mass design approach to the clustered tensegrity structures (CTS). By minimal mass, we mean, for a given clustered tensegrity structure subject to prescribed external loads, the minimal mass is achieved when all the structure members simultaneously fail (buckle or yield). This paper helps to find the lower bound of the required mass of the CTS. Firstly, we introduce the connectivity and clustering definitions through respective matrices and derive the nonlinear clustered tensegrity statics equation in terms of the nodal vector of the structure. Since the mass of each member is in a one-to-one correspondence to its force density (force by unit length), the static equilibrium equations of the CTS with respect to the force density and force vectors are also given. Then, we formulated the mass and gravity of hollow bars and strings subject to buckling and yielding conditions. Finally, a nonlinear optimization algorithm is presented to compute the minimal mass of any CTS with any given external force and topology. We also demonstrate that the traditional tensegrity structure (TTS) is a particular case of the CTS by defining the cluster matrix as an identity matrix. At last, numerical examples are given to validate the CTS minimal mass design approach. The method proposed by this research can be used for the lightweight design of tensegrity, truss, cable nets, membrane structures, etc.

Sequential Plastic Method for 2D frames limit analysis

This work focuses on the limit analysis or plastic calculation of slender beam planar frames using a new method. Sufficient static equilibrium equations are proposed by the Principle of Virtual Displacements (PVD) and kinematic compatibility equations are proposed by the Principle of Virtual Forces (PVF). The load level is increased and the structure is solved step by step until the plastic collapse. Equilibrium equations are posed by virtual problems in displacements and compatibility equations are posed by simple virtual problems in equilibrium. The method provides the following results: the collapse load factor, the final collapse mechanism, the bending moments and the accumulated rotations in the plastic hinges in each load step of the structure. This method has advantages over the classical methods; first, with respect to the step-by-step method based on matrix formulation, especially in the case of beams and/or columns with uniform distributed load, and secondly, with respect to the kinematic direct method, since the sequential method provides more information on the quantities involved in the plasticization process of the structure and also goes directly to the calculation of the collapse mechanism without the need to test or combine possible mechanisms.

Dynamic snap-through of bi-stable laminates with simply supported at four corners under impact loads

In this paper, nonlinear dynamic responses of bi-stable laminate being simply supported on four corner points, which is subjected to various impact loads, are analyzed from theoretical viewpoints. Consider a four layers asymmetric cross-ply bi-stable laminate composited by the carbon fiber reinforced resin matrix. A novel configuration function that satisfies the boundary condition is proposed to predict the equilibrium configuration. Then, employing the first order shear deformation theory, the von Kàrmàn strains and Hamilton principle, nonlinear dynamic equations of motion expressed by two curvatures are derived. To obtain the two stable configurations of the laminate, the inertia terms are ignored. With the aids of static equilibrium equation, the side length and curvature of the system are solved by using NLPSolve functions in the Maple software. The stable configurations are verified by comparing present results with those reported in available literatures. Consider the incremental load loading, the decreasing loading, the sinusoidal loading, the step loading, the triangular loading and periodic loading, the nonlinear dynamic response of the bi-stable laminate are discussed. The dynamic snap-through behavior, the intra-well oscillations and inter-wall oscillations are studied by numerical simulation in detail.

A Systematic survey on Estimation of Electrical Vehicle

Due to the gasoline crisis, electric vehicles are growing in popularity. The usage of electric and battery-powered automobiles is being encouraged worldwide. This campaign also heavily relies on the usage of renewable energy sources to provide electricity. In order to build electric vehicles, engineers use static and dynamic equations. Around the world, competitions are being held to design high-performance electric vehicles. The automotive industry and businesses are transitioning a portion of their fleet from gasoline-powered vehicles to batteryoperated electric vehicles. Most vehicles today offer good performance at slightly increased costs to the consumer. The battery technology is improving andhence enhancing the range of vehicles. The movement of use of electrical vehicle will strengthen as more people are getting involved in the design and use of electric vehicle.

On determination on some exact solutions of the stationary Navier-Stokes equations

In this paper, various exact solutions of the stationary Navier-Stokes equations, which describe the planar flow of an incompressible liquid (or gas), are determined, i.e., solutions containing the components of the velocity of flow - the functions u, v and the created pressure P. We mention that in the paper a series of exact solutions is obtained, in which the viscosity coefficient lambda participates explicitly.

Determining the Potential Function of the Stationary Vector Burgers’ Equation

We consider the inverse problem of determining the potential function of the stationary vector Burgers equations Δu – (u · ∇)u – q(x)u = 0. We give the well-posedness of the solution in H2 a small boundary value. Then, by linearization, we prove that the potential function can be determined from the boundary Cauchy data.

Multizonal Boundary and Internal Layers in the Singularly Perturbed Problems for a Stationary Equation of Reaction–Advection–Diffusion Type with Weak and Discontinuous Nonlinearity

Multizonal Boundary and Internal Layers in the Singularly Perturbed Problems for a Stationary Equation of Reaction–Advection–Diffusion Type with Weak and Discontinuous Nonlinearity

Structural Damage Identification Using the Optimal Achievable Displacement Variation.

To ensure the safe use of structures, it is essential to develop efficient damage identification techniques. In this paper, a brand-new approach to identifying structural deterioration based on static displacement is proposed. First, the relationship between the displacement variation and the damaged element is derived from the static response equations before and after damage. Subsequently, the optimal achievable displacement variation is defined to determine the damage location in the structure. A progressive elimination strategy is suggested to identify the real damaged parts and weed out the pseudo-damaged elements by measuring the distance between the measured and the best possible displacement variation. After determining the damage location, the corresponding damage extent can be calculated by a system of linear equations. The proposed approach has been tested on a beam structure and truss structure using simulated and experimental data. Compared with the existing static sensitivity method, the suggested method does not result in misjudgment and has higher identification accuracy. It has been demonstrated that the suggested approach is effective at locating and assessing the extent of structural damage.

Hemivariational inequality for contaminant reaction–diffusion model of recovered fracturing fluid in the wellbore of shale gas reservoir

The goal of this paper is to study the dynamic behavior of recovered fracturing fluid and the reaction–diffusion phenomenon of contaminants in the wellbore of shale gas reservoir during the early stage of fracturing fluid flowback. First, by applying various physical constitutive laws, a complicated recovered fracturing fluid model which consists of a stationary incompressible Stokes equation with a nonmonotone and multivalued friction law and a reaction–diffusion equation with the Neumann boundary condition, is established. Then, we use the variational techniques and the theory of multivalued analysis to formulate a new recovered fracturing fluid model. The latter is a variational system which consists of a hemivariational inequality with constraint driven by an elliptic variational equation. The system involves mutually coupled velocity and the concentration fields. Finally, under some mild assumptions, we employ a surjectivity theorem for pseudomonotone operators, monotonicity arguments and Schauder fixed point theorem to prove the existence of a weak solution to the recovered fracturing fluid model.

Utility of in silico prediction of target suppression for antibodies against soluble targets: static versus dynamic models.

Antibodies that bind soluble targets such as cytokines belong to an important class of immunotherapies. Target levels can significantly accumulate after antibody administration due to formation of antibody-target complex, accompanied with suppression in free target which is often difficult to measure. Being a surrogate for pharmacodynamic activity, free target suppression is often predicted using in silico tools. The objective of this work is to illustrate the utility of modelling and to compare static versus dynamic models in the prediction of free target suppression. Using binding principles, we have derived a static equation to predict free target suppression at steady state (FTSS). This equation operates with five input parameters and accounts for target accumulation over time. Its predictivity was compared to a dynamic model and to other existing metrics in literature via simulations and assumptions were illustrated. We demonstrated the utility of in silico tools in prediction of free target suppression using static and dynamic models and clarified the assumptions in key input parameters and their limitations. Predicted values using the FTSS equation correlate very well with those from the dynamic model at level > 20% target suppression, relevant for antagonistic antibodies. In silico tools are needed to predict target suppression by antibody drugs. Static or dynamic models can be used dependant on the scope, available data and undertaken assumptions. These tools can be used to guide discovery and development of antibodies and has the potential to reduce clinical failure.

Static and dynamic analyses of traditional Chinese timber columns under horizontal accelerations

Traditional Chinese timber columns are prone to rocking and sliding because the column feet are directly placed on foundation stones. Determining the motion of columns under horizontal accelerations is essential to preventing the collapse of traditional Chinese timber structures during earthquakes. This paper divides the states of motion of traditional Chinese timber columns into static and dynamic. For the static state, a coefficient describing the compression height is proposed based on the variation of the compressive zone at the column foot, and the moment-rotation relationship of the column foot is then established. For the dynamic state, the rocking and slide-rocking dynamic equations are proposed by considering the contribution of the column foot and the mortise-tenon joint to the lateral force of the timber columns. Finite element models of timber frames are constructed and validated through existing timber frame shaking table tests. Based on numerical simulation, this paper analyzed 125 groups of timber column models and 98 groups of timber frame models to validate the proposed static and dynamic equations. The results reveal that the proposed moment-rotation relationship of the column foot agreed with the simulated results at the initial stiffness and bearing capacity. Hence, the proposed dynamic equations reflect the timber columns' slide and rocking characteristics under horizontal acceleration. This investigation provides a theoretical base for the reinforcement and protection of traditional Chinese timber structures during earthquakes.

Pseudospectral Approach to the Shape Optimization of Beams Under Buckling Constraints

In this article, a direct transcription approach to the minimization of the volume of elastic straight beams undergoing plane deformation and subject to buckling loads is presented. In particular, the so-called pseudospectral method is employed, where state variables are approximated by Lagrange interpolating polynomials and static equations are collocated at Legendre-Gauss-Radau nonuniform mesh points. The resulting shape optimization problems are thus transcribed into constrained nonlinear programming problems, which in turn are solved by developed routines. Historical benchmark and academic problems such as simply supported beams subject to a concentrated compressing force, compressed and rotating cantilever beams and simply supported beams under a non-conservative follower distributed load are revisited and numerically solved under the conditions of plane deformation theory. Numerical solutions are discussed and compared to those obtained by the shooting method, which is largely employed for these problems, emphasizing how the proposed method could forecast optimal cross sectional area distributions within a unified fashion and without resorting to accurate guesses beforehand.

General Mathematical Model for Analysing the Bending Behaviour of Rectangular Concrete Beams with Steel, Fibre-Reinforced Polymers (FRP) and Hybrid FRP–Steel Reinforcements

The design guidelines available in building codes for steel and fibre reinforced polymer (FRP) reinforced concrete (RC) beams have been developed on the basis of empirical models. While these models are successfully used for practical purposes, they require continuous improvements with more experimental data. This paper aims to develop a general mathematical model derived from the intrinsic material properties of concrete and certain reinforcements to analyse the bending behaviour of reinforced concrete beams. The proposed model takes into account the effects of non-linearity and ductility on the real behaviour of concrete under compression as well as the concrete tension stiffening. The model focused on analysing the flexural behaviour of rectangular steel, FRP and hybrid FRP–steel RC beams, using the moment–curvature relationship. A general static equilibrium equation was developed and mathematically solved with precise methods to establish a moment–curvature relationship. The effective flexural stiffness (EFS) is therefore calculated by the slope of the moment–curvature diagram, and then the load–deflection response is immediately deduced according to the loading conditions. The present model results were compared with numerous test data reported by various researchers. The comparisons reveal a good accuracy for predicting the EFS and load–deflection response for either steel, FRP, and hybrid reinforced concrete beams, with an error average less than 10%.

Note on rotating BEC under a confining potential

Recently there have been analytical studies concerning the threshold on the instability for a rotating BEC under a trapping potential. In this note, we obtain a critical lower bound μ∗ that ensures the existence of ground state solutions for the L2-critical RNLS in 2d and 3d, where an anisotropic harmonic potential is present. In addition, we perform numerical estimation for the corresponding stationary equation for the ground state and the vortex state solutions. Our findings indicate agreement on the sharp bound from both analytical and numerical demonstration. Among others we prove a blowup alternative result for RNLS with an inhomogeneous nonlinearity.

Applications of Zvonkin’s Transform to Stationary Kolmogorov Equations

—In this note we develop a new analytic version of Zvonkin’s transform of the drift coefficient of a stationary Kolmogorov equation and apply this transform to derive the Harnack inequality for nonnegative solutions in the case where the diffusion matrix is not locally Sobolev. We also obtain a generalization of the known theorem of Hasminskii on existence of a probability solution to the stationary Kolmogorov equation.