Weak Coercivity Research Articles

FEM simulation of thixo‐viscoplastic flow problems: Error analysis

AbstractThis note is concerned with the essential part of Finite Element Methods (FEM) approximation of error analysis for quasi‐Newtonian modelling of thixo‐viscoplastic (TVP) flow problems. The developed FEM settings for thixotropic generalized Navier‐Stokes equations is based on a constrained monotonicity and continuity for the coupled system, which is a cornerstone for an efficient monolithic Newton‐multigrid solver. The manifested coarseness in the energy inequality by means of proportional dependency of its constants on regularization, nonoptimal estimate for microstructure, and extra regularity requirement for velocity, is due to the weak coercivity of microstructure operator on one hand and the modelling approach on the other hand, which we dealt with stabilized higher order FEM. Furthermore, we show the importance of taking into consideration the thixotropy inhabited in material by presenting the numerical solutions of TVP flow problems in a 4:1 contraction configuration.

Brezis pseudomonotone bifunctions and quasi equilibrium problems via penalization

In this paper we investigate quasi equilibrium problems in a real Banach space under the assumption of Brezis pseudomonotonicity of the function involved. To establish existence results under weak coercivity conditions we replace the quasi equilibrium problem with a sequence of penalized usual equilibrium problems. To deal with the non compact framework, we apply a regularized version of the penalty method. The particular case of set-valued quasi variational inequalities is also considered.

A Generalized Ky Fan Minimax Inequality on Finite-Dimensional Spaces

An existence result for a generalized inequality over a possible unbounded domain in a finite-dimensional space is established. The proof technique allows to avoid any monotonicity assumption. We adapt a weak coercivity condition introduced in Castellani and Giuli (J Glob Optim 75:163–176, 2019) for a generalized game which extends an older one proposed by Konnov and Dyabilkin (J Glob Optim 49:575–577, 2011) for equilibrium problems. Our main result encompasses and generalizes several existence results for equilibrium, quasiequilibrium and fixed-point problems.

Evolution from sinusoidal to collinear A-type antiferromagnetic spin-ordered magnetic phase transition in Tb1−x Pr x MnO3 solid solution

The present study reports on the structural and magnetic phase transitions in Pr-doped polycrystalline Tb0.6Pr0.4MnO3, using high-resolution neutron powder diffraction (NPD) collected at SINQ spallation source, to emphasize the suppression of the sinusoidal magnetic structure of pure TbMnO3 and the evolution to a collinear A-type antiferromagnetic ordering. The phase purity, Jahn–Teller distortion, and one-electron bandwidth for eg orbital of Mn3+ cation have been calculated for polycrystalline Tb0.6Pr0.4MnO3, in comparison to the parent materials TbMnO3 and PrMnO3, through the Rietveld refinement study from x-ray diffraction data at room temperature, which reveals the GdFeO3 type orthorhombic structure of Tb0.6Pr0.4MnO3 having Pnma space group symmetry. The temperature-dependent zero field-cooled and field-cooled dc magnetization study at low temperature down to 5 K reveals a variation in the magnetic phase transition due to the effect of Pr3+ substitution at the Tb3+ site, which gives the signature of the antiferromagnetic nature of the sample, with a weak ferromagnetic component at low temperature-induced by an external magnetic field. The field-dependent magnetization study at low temperatures gives the weak coercivity having the order of 2 kOe, which is expected due to the canted-spin arrangement or ferromagnetic nature of Terbium ordering. The NPD data for Tb0.6Pr0.4MnO3 confirms that the nuclear structure of the synthesized sample maintains its orthorhombic symmetry down to 1.5 K. Also, the magnetic structures have been solved at 50 K, 25 K, and 1.5 K through the NPD study, which shows an A-type antiferromagnetic spin arrangement having the magnetic space group Pn′ma′.

Assessment of Magnetic Techniques for Understanding Complex Mixtures of Magnetite and Hematite: The Inuyama Red Chert

AbstractMagnetite and hematite mixtures occur widely in nature. Magnetic unmixing of the signals recorded by these minerals can be important for assessing the origin of their respective paleomagnetic remanences and for extracting geological and paleoenvironmental information. However, unmixing magnetic signals from complex magnetite and hematite mixtures is difficult because of the weak magnetization and high coercivity of hematite. We assess here the relative effectiveness of first‐order reversal curve (FORC) and extended FORC‐type diagrams, FORC‐principal component analysis (PCA), isothermal remanent magnetization (IRM) curve decomposition, and PCA of remanent hysteretic curves for unmixing magnetic components in samples from the magnetically complex Inuyama red chert, Japan. We also further characterize the domain state and coercivity distributions of both magnetite and hematite with FORC‐PCA and IRM acquisition analysis in the red chert. We show that IRM curve decomposition can provide valuable component‐specific information linked to coercivity, while FORC‐PCA enables effective magnetic domain state identification. PCA of remanent hysteretic curves provides useful information about the most significant factors influencing remanence variations and subtle coercivity changes. To identify components in complex magnetite and hematite mixtures, we recommend PCA analysis of remanent hysteretic curves combined with FORC analysis of representative samples to identify domain states and coercivity distributions.

Well-Posedness and Exponential Estimates for the Solutions to Neutral Stochastic Functional Differential Equations with Infinite Delay

AbstractIn this work, neutral stochastic functional differential equations with infinite delay (NSFD-EwID) have been addressed. By using the Euler-Maruyama scheme and a localization argument, the existence and uniqueness of solutions to NSFDEwID at the state space Cr under the local weak monotone condition, the weak coercivity condition and the global condition on the neutral term have been investigated. In addition, the L2 and exponential estimates of NSFDEwID have been studied.

Optimal-order finite element approximations to variable-coefficient two-sided space-fractional advection-reaction-diffusion equations in three space dimensions

We analyze finite element approximations to variable-coefficient two-sided space-fractional advection-reaction-diffusion equations in three space dimensions. It is known that the corresponding bilinear form may lose coercivity due to the variable diffusivity coefficient. We prove the weak coercivity of the bilinear form via the Gårding's inequality and thus derive the optimal-order error estimate of the finite element method. The developed method is further extended to analyze the time-dependent problems. Numerical experiments are given to validate the theoretical findings.

Decomposable penalty method for generalized game problems with joint constraints

ABSTRACT We consider an extension of a non-cooperative game problem where players have joint binding constraints. In this case, justification of a generalized equilibrium point needs a reasonable mechanism for attaining this state. We combine a penalty method and shares allocation of right-hand sides, which replaces the initial problem with a sequence of the usual Nash equilibrium problems together with an upper level variational inequality as a master problem. In order to obtain a completely decomposable problem at the lower level, we apply its additional equivalent transformation. Convergence of solutions of these auxiliary penalized problems to a solution of the initial game problem is established under weak coercivity conditions.

Magnetization spin reversal and neutron diffraction study of polycrystalline Tb0.55Sr0.45MnO3

We have investigated the structural and magnetic phase transitions in Sr doped polycrystalline Tb0.55Sr0.45MnO3, using temperature-dependent high-resolution neutron powder diffraction in the HRPT diffractometer at PSI to address the origin of magnetization reversal at low temperature. The solid solution Tb0.55Sr0.45MnO3 crystallizes in O′ type orthorhombic structure having the Pnma symmetry. The substitution of the divalent cation of Sr2+ at the site of Tb3+ dilutes the Mn–Tb interaction and affects the magnetic structure, which was observed by the field and temperature-dependent dc magnetization and neutron diffraction study. The temperature-dependent zero field-cooled and field cooled dc magnetization study reveals the indication of spin reversal phenomena, which is the nature of canted antiferromagnetic or ferrimagnetic transition at 100 Oe because of Mn–Mn sub-lattices interaction below 65 K. By increasing the applied magnetic field up to 20 kOe, a weak ferromagnetic type of behaviour is observed. The field-dependent magnetization shows weak coercivity due to the canted spin structure at low temperatures. The low-temperature neutron diffraction study of polycrystalline Tb0.55Sr0.45MnO3 reveals the non-collinear canted antiferromagnetic structure due to Tb ordering at 1.5 K, which turns into a non-collinear ferromagnetic structure along with spin canting followed by the spin reversal phenomena at 25 K.

On the Existence of Solutions and Tikhonov Regularization of Hemivariational Inequality Problems

This paper is devoted to the existence of solution and Tikhonov regularization theory for a class of hemivariational inequalities. We give the existence of solutions for the class of hemivariational inequalities when the mappings satisfy the so-called hemivariational inequality property and a rather weak coercivity condition. The existence result allows us to deduce the Tikhonov regularization result. Our results generalize some results by He (Abstr. Appl. Anal. 2012, 172061, 2012) and others.

A coercivity condition for nonmonotone quasiequilibria on finite-dimensional spaces

New existence results for quasiequilibrium problems on unbounded feasible sets in a finite-dimensional space and without any assumption of monotonicity are established. The key point behind these results is a weak coercivity condition for a generalized game which extends a recent one proposed in Konnov and Dyabilkin (J Glob Optim 49:575–587, 2011) for equilibrium problems and an older one given in Cubiotti (Comput Math Appl 30:11–22, 1995) for quasiequilibrium problems. Some examples are also given.

Equilibrium formulations of relative optimization problems

We consider relative or subjective optimization problems where the goal function and feasible set are dependent of the current state of the system under consideration. We propose equilibrium formulations of the corresponding problems that lead to general (quasi-)equilibrium problems. We propose to apply a regularized version of the penalty method for the general quasi-equilibrium problem, which enables us to establish existence results under weak coercivity conditions and replace the quasi-equilibrium problem with a sequence of the usual equilibrium problems. We describe several examples of applications and show that the subjective approach can be extended to non-cooperative game problems.

Large magnetostriction in chemically fabricated CoFe2O4 nanoparticles and its temperature dependence

Cobalt ferrite nanoparticles were synthetized by co-precipitation technique from nitrate precursors. X-ray diffraction confirms the formation of the spinel ferrite. SEM and HRTEM micrographs corroborate the formation of agglomerated nanoparticles of 8–35 nm size range with 21.7 nm average size. The broad size distribution and considerable agglomeration of the particles hamper the superparamagnetic state of the sample, provoking a weak coercivity at room temperature. In contrast to several other magnetic nanostructures, the ZFC-FC curves of cobalt-ferrite nanoparticles revealed an unusually large magnetostriction, evidenced by larger values of magnetization in their ZFC curves than the FC curves in the measured temperature range 20–375 K. The behavior could be reversed on applying high magnetic fields. However, the maximum of ZFC curve reduces on increasing the applied magnetic field.

Effect of local elasticity of the matrix on magnetization loops of hybrid magnetic elastomers

To model magnetization loops of magnetorheological elastomers (MREs) with magnetically hard filler, we consider an assembly of single-domain particles possessing inversion-symmetrical shape and embedded in a soft polymer matrix. To describe the intrinsic behavior of the particle magnetic moments under an applied field, the Stoner-Wohlfarth approach is employed. Unlike the case of solid matrix, the particle in a MRE is able to rotate relative to its elastic environment, so that its equilibrium orientation results from the balance between the magnetic torque (exerted by the applied field at any magnetic moment that does not point along the field) and the elastic torque generated by the matrix. We assume that elastic resistance to the field-induced particle rotation could be presented as comprising of two contributions. The first one is valid for any not perfectly spherical particle and is independent of the particle–matrix adhesion. It reflects the fact that, when trying to rotate, the particle has to “shoulder its way” by deforming the adjoining regions of the matrix. In that case, for angular deviations up to 90° from the initial position, the resistance torque increases. However, as soon as the rotation angle grows up to the value but infinitesimally exceeding 90°, the elastic torque changes its sign and from now on forces the particle to rotate to 180°, where it attains the geometrical position that coincides with the initial one. Evidently, this process is of the barrier type: both orientations of the particle are equal in elastic energy. The second mechanism stems from the “memory” that a given particle has of its initial state and may be caused, for example, by some macromolecules grafted to its surface while curing the matrix of the MRE. This restoring force always tends to drag the particle to its 0° (initial) position. Therefore, the observed magnetization of a MRE sample comes out as a result of joint interplay of the Stoner-Wohlfarth and the two elastic mechanisms. In our simulations, we show that the proposed model in a natural way accounts for the two essential features observed (solely or together) in experiment, namely: (i) weak net coercivity of the MREs, whose filler particles as themselves are highly coercive, and (ii) asymmetric positions of the magnetization loops with respect to H=0 point at the (M-H) plane.

Regularized penalty method for non-stationary set valued equilibrium problems in Banach spaces

In this research works, we consider the general regularized penalty method for non-stationary set valued equilibrium problem in a Banach space. We define weak coercivity conditions and show that the weak and strong convergence problems of the regularized penalty method.

Existence results for -quasimonotone equilibrium problems in convex metric spaces

This paper deals with equilibrium problems in the setting of metric spaces with a continuous convex structure. We extend Fan’s 1984 KKM theorem to convex metric spaces in order to employ some weak coercivity conditions to establish existence results for suitable local Minty equilibrium problems, where the involved bifunctions are -quasimonotone. By an approach which is based on the concept of the strong -sign property for bifunctions, we obtain existence results for equilibrium problems which generalize some results in the literature.

An experimental and theoretical study of the optical, electronic, and magnetic properties of novel inverted α-Cr2O3@α-Mn0.35Cr1.65O2.94core shell nanoparticles

Abstract

Least-Squares Proper Generalized Decompositions for Weakly Coercive Elliptic Problems

Proper generalized decompositions (PGDs) are a family of methods for efficiently solving high-dimensional PDEs, which seek to find a low-rank approximation to the solution of the PDE a priori. Convergence of PGD algorithms can only be proven for problems which are continuous, symmetric, and strongly coercive. In the particular case of problems which are only weakly coercive we have the additional issue that weak coercivity estimates are not guaranteed to be inherited by the low-rank PGD approximation. This can cause stability issues when employing a Galerkin PGD approximation of weakly coercive problems. In this paper we propose the use of PGD algorithms based on least-squares formulations which always lead to symmetric and strongly coercive problems and hence provide stable and provably convergent algorithms. Taking the Stokes problem as a prototypical example of a weakly coercive problem, we develop and compare rigorous least-squares PGD algorithms based on continuous least-squares estimates for two dif...

Ekeland’s principle for cyclically antimonotone equilibrium problems

In this paper, we deal with equilibrium problems without convexity assumptions either for the domain or for the function involved. First we give an Ekeland’s variational principle for equilibrium problems on complete metric spaces which generalizes some recent results. The main improvement consists in widening the class of bifunctions for which the variational principle holds: instead of a triangle inequality property a suitable approximation from below of the bifunction f is required and we show that this condition is equivalent to the cyclic antimonotonicity of f. We furnish a simple and direct proof of the existence of equilibria for such equilibrium problems in presence of a compact feasible set without passing through the existence of approximate solutions. This fact allows us to remove the metric structure on the topological space and additional technical conditions. Subsequently, a weak coercivity condition is introduced to deal with the non weakly compact case in reflexive Banach space. Finally, applications both to uncountable systems of equilibrium problems and to the quasiequilibrium problems are also discussed.

An Unconditionally Stable Discontinuous Galerkin Method for the Elastic Helmholtz Equations with Large Frequency

In this paper we propose and analyze an interior penalty discontinuous Galerkin (IP-DG) method using piecewise linear polynomials for the elastic Helmholtz equations with the first order absorbing boundary condition. It is proved that the sesquilinear form for the problem satisfies a generalized weak coercivity property, which immediately infers a stability estimate for the solution of the differential problem in all frequency regimes. It is also proved that the proposed IP-DG method is unconditionally stable with respect to both frequency $$\omega $$ź and mesh size h. Sub-optimal order (with respect to h) error estimates in the broken $$H^1$$H1-norm and in the $$L^2$$L2-norm are obtained in all mesh regimes. These estimate improve to optimal order when the mesh size h is restricted to the pre-asymptotic regime (i.e., $$\omega ^{\beta } h =O(1)$$źβh=O(1) for some $$1\le \beta <2$$1≤β<2). Numerical experiments are also presented to validate the theoretical results and to numerically examine the pollution effect (with respect to $$\omega $$ź) in the error bounds.