Additional Compatibility Conditions Research Articles

An accurate and numerically efficient time-varying mesh stiffness model for modified straight bevel gear under load

Gear time-varying mesh stiffness (TVMS) plays an important role in the analysis accuracy of a gear transmission mechanism. In previous research, the TVMS of a modified straight bevel gear (SBG) was mainly obtained by the finite element method (FEM), whose computation time is usually long to ensure computational accuracy. By contrast, this work proposes a fast calculation method with high accuracy for the TVMS of a modified SBG. First, B-spline surfaces are introduced to reconstruct a modified SBG using discrete tooth surface points. Next, based on the tooth contact analysis and loaded tooth contact analysis, combined with energy conservation principle and infinitesimal method, a semi-analytical instantaneous single mesh stiffness (SMS) model for the modified SBG is derived. Then, using the additional deformation compatibility condition and principle of force balance, the calculation method of the TVMS of the modified SBG is obtained. Finally, a comparison of the mesh stiffness calculation results between the method of this study and that of FEM is completed. The results show that the semi-analytical TVMS model used in this study can be accurately and quickly solved.

Boundary behavior for the heat equation on the half‐line

The initial‐boundary value problem for the heat equation on with nonzero Dirichlet boundary data is studied rigorously, with emphasis on the behavior of the solution along the boundary of the spatiotemporal domain, that is, in the limits or , including the case . Our results specify how the regularity of the solution at the boundary is dictated by the level of smoothness of the relevant initial and boundary data. To the best of our knowledge, for unbounded domains like the half‐line , such results are not available in the literature. The corresponding analysis for the derivatives of the solution is also performed and yields additional compatibility conditions on the initial and boundary data which guarantee the extension of the solution to a function up to the boundary, that is, on . The analysis relies on the explicit solution formula provided by the unified transform, also known as the Fokas method, whose rigorous derivation is presented. The main advantage of this formula, namely, its uniform convergence at the boundary, largely reduces the proof of our results to fundamental tools from real and complex analysis such as Lebesgue's dominated convergence theorem and Cauchy's theorem. Moreover, thanks to uniform convergence, the unified transform solution formula can actually be directly evaluated at the boundary to yield the prescribed initial and boundary data.

Stabilised Variational Multi-scale Finite Element Formulations for Viscoelastic Fluids

The objective of this article is to summarise the work that we have been doing as a group in the context of stabilised finite element formulations for viscoelastic fluid flows. Viscoelastic fluids are complex non-Newtonian fluids, characterised by having an irreducible constitutive equation that needs to be solved coupled with the momentum and continuity equations. The finite element approximation of this kind of fluids presents several numerical difficulties. It inherits obviously the problems associated with the approximation of the incompressible Navier–Stokes equations. But, on top of that, now the constitutive equation is highly non-linear, with an advective term that may lead to both global and local oscillations in the numerical approximation. Moreover, even in the case of smooth solutions, it is necessary to meet some additional compatibility conditions between the velocity and the stress interpolation in order to ensure control over velocity gradients. The stabilised methods detailed in this work allow one to use equal order or even arbitrary interpolation for the problem unknowns ( $${\varvec{\sigma }}$$ - $${\varvec{u}}$$ -p) (elastic deviatoric stress-velocity-pressure) and to stabilise dominant convective terms, and all of them can be framed in the context of variational multi-scale methods. Some additional numerical ingredients that are introduced in this article are the treatment of the non-linearities associated with the problem and the possibility to introduce a discontinuity-capturing technique to prevent local oscillations. Concerning the constitutive equation, both the standard as the logarithmic conformation reformulation are discussed for stationary and time-dependent problems, and different versions of stabilised finite element formulations are presented in both cases.

Local well-posedness to the three-dimensional barotropic compressible magnetohydrodynamic equations with vacuum

This paper considers the local well-posedness of strong solutions to the three-dimensional barotropic compressible magnetohydrodynamic equations with the initial density containing vacuum. We first derive the local existence and uniqueness of strong solutions. Then, we show the continuous dependence of strong solutions on the initial data under an additional compatibility condition.

On local strong and classical solutions to the three-dimensional barotropic compressible Navier-Stokes equations with vacuum

We consider the local well-posedness of strong and classical solutions to the three-dimensional barotropic compressible Navier-Stokes equations with density containing vacuum initially. We first prove the local existence and uniqueness of the strong solutions, where the initial compatibility condition proposed by Cho et al. (2004), Cho and Kim (2006) and Choe and Kim (2003) is removed in a suitable sense. Then, the continuous dependence of strong solutions on the initial data is derived under an additional compatibility condition. Moreover, for the initial data satisfying some additional regularity and the compatibility condition, the strong solution is proved to be a classical one.

The Gap Between Linear Elasticity and the Variational Limit of Finite Elasticity in Pure Traction Problems

A limit elastic energy for pure traction problem is derived from re-scaled nonlinear energy of an hyperelastic material body subject to an equilibrated force field. We show that the strains of minimizing sequences associated to re-scaled non linear energies weakly converge, up to subsequences, to the strains of minimizers of a limit energy, provided an additional compatibility condition is fulfilled by the force field. The limit energy is different from classical energy of linear elasticity; nevertheless the compatibility condition entails the coincidence of related minima and minimizers. A strong violation of this condition provides a limit energy which is unbounded from below, while a mild violation may produce a limit energy with infinitely many extra minimizers which are not minimizers of standard linear elastic energy and whose strains are not uniformly bounded. A relevant consequence of this analysis is that a rigorous validation of linear elasticity fails for compressive force fields that do not fulfil such compatibility condition.

On local strong solutions to the three-dimensional nonhomogeneous Navier–Stokes equations with density-dependent viscosity and vacuum

We study the three-dimensional nonhomogeneous Navier–Stokes equations with density-dependent viscosity and vacuum on Ω⊂R3, which is either a bounded domain or a usual unbounded one such as the whole space R3 and an exterior one. In particular, the initial density can have compact support when Ω is unbounded. For initial data without additional compatibility conditions, we prove that there exists a unique local strong solution to the initial and initial boundary value problems. Moreover, the continuous of strong solutions on the initial data is derived under an additional compatibility condition.

On local strong solutions to the three-dimensional nonhomogeneous incompressible magnetohydrodynamic equations with density-dependent viscosity and vacuum

This paper concerns the three-dimensional nonhomogeneous incompressible magnetohydrodynamic equations with density-dependent viscosity and vacuum on $$\varOmega \subset \mathbb {R}^3$$ . The domain $$\varOmega \subset \mathbb {R}^3$$ is a general connected smooth one, either bounded or unbounded. In particular, the initial density can have compact support when $$\varOmega $$ is unbounded. First, we obtain the local existence and uniqueness of strong solution to the three-dimensional nonhomogeneous incompressible magnetohydrodynamic equations without any compatibility condition assumed on the initial data. Then, we also prove the continuous dependence of strong solution on the initial data under an additional compatibility condition.

A Class of Control Certificates to Ensure Reach-While-Stay for Switched Systems

In this article, we consider the problem of synthesizing switching controllers for temporal properties through the composition of simple primitive reach-while-stay (RWS) properties. Reach-while-stay properties specify that the system states starting from an initial set I, must reach a goal (target) set G in finite time, while remaining inside a safe set S. Our approach synthesizes switched controllers that select between finitely many modes to satisfy the given RWS specification. To do so, we consider control certificates, which are Lyapunov-like functions that represent control strategies to achieve the desired specification. However, for RWS problems, a control Lyapunov-like function is often hard to synthesize in a simple polynomial form. Therefore, we combine control barrier and Lyapunov functions with an additional compatibility condition between them. Using this approach, the controller synthesis problem reduces to one of solving quantified nonlinear constrained problems that are handled using a combination of SMT solvers. The synthesis of controllers is demonstrated through a set of interesting numerical examples drawn from the related work, and compared with the state-of-the-art tool SCOTS. Our evaluation suggests that our approach is computationally feasible, and adds to the growing body of formal approaches to controller synthesis.

On classical solutions to the Cauchy problem of the 2D compressible non-resistive MHD equations with vacuum states

In this paper, we investigate the Cauchy problem of the compressible non-resistive MHD on with a vacuum as the far field density. We prove that the two-dimensional (2D) Cauchy problem has a unique local strong solution provided that the initial density and magnetic field decay are not too slow at infinity. Furthermore, if the initial data satisfies some additional regularity and compatibility conditions, the strong solution becomes a classical one. Additionally, we establish a blowup criterion for the 2D compressible non-resistive MHD depending solely on the density and magnetic fields.

Well-posedness theory for degenerate parabolic equations on Riemannian manifolds

We consider the degenerate parabolic equation∂tu+divfx(u)=div(div(Ax(u))),x∈M,t≥0 on a smooth, compact, d-dimensional Riemannian manifold (M,g). Here, for each u∈R, x↦fx(u) is a vector field and x↦Ax(u) is a (1,1)-tensor field on M such that u↦〈Ax(u)ξ,ξ〉, ξ∈TxM, is non-decreasing with respect to u. The fact that the notion of divergence appearing in the equation depends on the metric g requires revisiting the standard entropy admissibility concept. We derive it under an additional geometry compatibility condition and, as a corollary, we introduce the kinetic formulation of the equation on the manifold. Using this concept, we prove well-posedness of the corresponding Cauchy problem.

Noncommutative reproducing kernel Hilbert spaces

The theory of positive kernels and associated reproducing kernel Hilbert spaces, especially in the setting of holomorphic functions, has been an important tool for the last several decades in a number of areas of complex analysis and operator theory. An interesting generalization of holomorphic functions, namely free noncommutative functions (e.g., functions of square-matrix arguments of arbitrary size satisfying additional natural compatibility conditions), is now an active area of research, with motivation and applications from a variety of areas (e.g., noncommutative functional calculus, free probability, and optimization theory in linear systems engineering). The purpose of this article is to develop a theory of positive kernels and associated reproducing kernel Hilbert spaces for the setting of free noncommutative function theory.

Partially ordered metric spaces produced by T0-quasi-metrics

We continue our investigation on those (partially) ordered metric spaces (X,m,≤) for which there exists a T0-quasi-metric d on X such that sup⁡{d,d−1}=m and for any x,y∈X we have that x≤y if and only if d(x,y)=0. In particular such partially ordered metric spaces are determined (in the sense of Nachbin) by a T0-quasi-uniformity with a countable base.We show that the compatibility conditions obtained in an earlier study between partial order and metric on the investigated spaces X become more transparent when additional compatibility conditions between metric and partial order with appropriate algebraic operations on X are assumed to hold.

단부 증타 보강된 RC 전단벽체의 전단강도

내진 보강 공법의 보강효과 파악을 위해선 이에 대한 이론적 규명이 필요하다. 이에 본 연구는 단부에 기둥 부재를 증타한 단근 배근된 전단벽체를 대상으로 해 보강되는 기둥의 상세와 일체화 거동을 고려한 전단강도 모델을 제안했다. 이 모델은 증타된 기둥부재의 전단변형이 집중되는 길이를 가정해 이를 일체화된 벽체 내에 발생하는 스트럿 두께 산정에 이용했다. 뿐만 아니라 이 길이 내에 집중 발생하는 전단변형률을 유도, 이를 일체화 거동을 고려한 적합성 조건을 기존 전단벽체의 해석 알고리즘에 도입함으로써 증타 보강된 벽체의 해석 알고리즘을 새로이 제안했다. 또한 제안된 알고리즘을 통해 계산된 전단강도로 횡력 저항 메커니즘을 단일 스트럿 화 시켜줌으로써 보강부재의 초기강성을 제안하였다. 본 연구에서 제안한 방법의 타당성을 확인하기 위해 해석 결과 값을 본 연구에서 수행한 증타 보강된 벽체를 대상으로 한 실험뿐만 아니라 기존 RC 골조 내에 RC 전단벽체를 신설해 보강한 실험의 결과와 비교해 본 결과, 기둥부재의 상세를 고려하면서 동시에 기존 기준들에 비해 실험결과에 가깝게 예측할 수 있음을 확인했다.

Global regularity to the 3D incompressible MHD equations

In this paper, we prove the global regularity to the 3D nonhomogeneous incompressible magnetohydrodynamic equations. Let ϱ0, m0 and H0 be the initial density, momentum and magnetic field, respectively. We establish a unique strong solution on R3×(0,T) for any 0<T<∞ under the assumption that the viscosity coefficient μ>0 is sufficiently large, or ‖|m0|2/ϱ0‖L12+‖H0‖L22 or ‖∇u0‖L22+‖H0‖H12 is small enough. Moreover, if the given data are more regular and satisfy an additional compatibility condition for the existence of strong solution, then we show that the strong solution is indeed a classical one. Moreover, the weak–strong uniqueness of solutions is also obtained, which generalizes the result in [13] to the case of vacuum.

On the Global Motion of Viscous Compressible Barotropic Flows Subject to Large External Potential Forces and Vacuum

We study the Cauchy problem of three-dimensional compressible barotropic Navier--Stokes equations with large external potential forces. We first prove that the Cauchy problem has a unique global strong solution with large oscillations and interior vacuum, provided the initial data are of small energy and the unique steady state is strictly away from vacuum. Next, we show that if the initial data are more regular and satisfy some additional compatibility conditions, the strong solution is indeed a classical one away from the initial time. Finally, as by-products, the global existence and large-time behavior of weak solutions with large external potential forces and discontinuous initial data containing vacuum are also obtained.

Global regularity of the three-dimensional equations for nonhomogeneous incompressible fluids

This paper is concerned with the global well-posedness of strong and classical solutions for the three-dimensional nonhomogeneous incompressible Navier–Stokes equations subject to vacuum and external forces. Let ϱ0,m0 and f be the initial density, initial momentum and potential external force, respectively. We first show that there exists a global strong solution (ϱ,u) on R3×(0,T) for any 0<T<∞, provided that the viscosity coefficient μ>0 is sufficiently large, or ‖ϱ0‖L∞ or ‖|m0|2/ϱ0‖L1+‖ϱ0‖L2‖f‖L2 or ‖∇(m0/ϱ0)‖L2+‖∇f‖H1 is small enough. Although the density may vanish in some open sets, it is only assumed that u0≜m0/ϱ0 is well defined and satisfies (ϱ01/2u0,∇u0)∈L2. A uniqueness result is also proved. Next, if the given data are more regular and satisfy an additional compatibility condition used in Choe and Kim (2003) for the existence of strong solution, then the strong solution is indeed a classical one.

On local classical solutions to the Cauchy problem of the two-dimensional barotropic compressible Navier–Stokes equations with vacuum

This paper concerns the Cauchy problem of the barotropic compressible Navier–Stokes equations on the whole two-dimensional space with vacuum as far field density. In particular, the initial density can have compact support. When the shear and the bulk viscosities are a positive constant and a power function of the density respectively, it is proved that the two-dimensional Cauchy problem of the compressible Navier–Stokes equations admits a unique local strong solution provided the initial density decays not too slow at infinity. Moreover, if the initial data satisfy some additional regularity and compatibility conditions, the strong solution becomes a classical one.

On Rational Pairings of Functors

In the theory of coalgebras C over a ring R, the rational functor relates the category $_{C^*}{\mathbb{M}}$ of modules over the algebra C * (with convolution product) with the category $^C{\mathbb{M}}$ of comodules over C. This is based on the pairing of the algebra C * with the coalgebra C provided by the evaluation map ${\rm ev}:C^*\otimes_R C\to R$ . The (rationality) condition under consideration ensures that $^C{\mathbb{M}}$ becomes a coreflective full subcategory of $_{C^*}{\mathbb{M}}$ . We generalise this situation by defining a pairing between endofunctors T and G on any category ${\mathbb{A}}$ as a map, natural in $a,b\in {\mathbb{A}}$ , $$ \beta_{a,b}:{\mathbb{A}}(a, G(b)) \to {\mathbb{A}}(T(a),b), $$ and we call it rational if these all are injective. In case T = (T, m T , e T ) is a monad and G = (G, δ G , e G ) is a comonad on ${\mathbb{A}}$ , additional compatibility conditions are imposed on a pairing between T and G. If such a pairing is given and is rational, and T has a right adjoint monad T ⋄ , we construct a rational functor as the functor-part of an idempotent comonad on the T-modules ${\mathbb{A}}_{T}$ which generalises the crucial properties of the rational functor for coalgebras. As a special case we consider pairings on monoidal categories.

Uniqueness and existence of spirals moving by forced mean curvature motion

In this paper, we study the motion of spirals by mean curvature type motion in 1 the (two dimensional) plane. Our motivation comes from dislocation dynamics; in this context, spirals appear when a screw dislocation line reaches the surface of a crystal. The first main result of this paper is a comparison principle for the corresponding parabolic quasi-linear equation. As far as motion of spirals are concerned, the novelty and originality of our setting and results come from the fact that, first, the singularity generated by the attached end point of spirals is taken into account for the first time, and second, spirals are studied in the whole space. Our second main result states that the Cauchy problem is well-posed in the class of sub-linear weak (viscosity) solutions. We also explain how to get the existence of smooth solutions when initial data satisfy an additional compatibility condition.