Bounds and estimates on the effective properties for nonlinear composites

In this paper we derive lower bounds and upper bounds on the effective properties for nonlinear heterogeneous systems. The key result to obtain these bounds is to derive a variational principle, which generalizes the variational principle by P. Ponte Castaneda from 1992. In general, when the Ponte Castaneda variational principle is used one only gets either a lower or an upper bound depending on the growth conditions. In this paper we overcome this problem by using our new variational principle together with the bounds presented by Lukkassen, Persson and Wall in 1995. Moreover, we also present some examples where the bounds are so tight that they may be used as a good estimate of the effective behavior.     

Estimates for the density of a nonlinear Landau process

The aim of this paper is to obtain estimates for the density of the law of a specific nonlinear diffusion process at any positive bounded time. This process is issued from kinetic theory and is called Landau process, by analogy with the associated deterministic Fokker-Planck-Landau equation. It is not Markovian, its coefficients are not bounded and the diffusion matrix is degenerate. Nevertheless, the specific form of the diffusion matrix and the nonlinearity imply the non-degeneracy of the Malliavin matrix and then the existence and smoothness of the density. In order to obtain a lower bound for the density, the known results do not apply. However, our approach follows the main idea consisting in discretizing the interval time and developing a recursive method. To this aim, we prove and use refined results on conditional Malliavin calculus. The lower bound implies the positivity of the solution of the Landau equation, and partially answers to an analytical conjecture. We also obtain an upper bound for the density, which again leads to an unusual estimate due to the bad behavior of the coefficients.     

An improved lower bound for deflections of laminated cantilever beams including the effect of transverse shear deformation

A previous paper (reference [3]) has formulated a version of the well-known plane stress problem of the end-loaded cantilever beam of narrow rectangular cross section in a way which allowed the statement of upper and lower bounds for the deflection of the beam, through use of the principles of minimum potential and complementary energy. In extension of the work in reference [3] we here derive an improved lower bound, with the consequence that now the upper and the lower bound coincide for the limiting case of vanishing depth to span ratio of the beam. This means that we have a rigorous proof of the correctness of the ‘elementary strength of materials’ result for the case of a very slender beam, in addition to quantitatively reliable corrections to the elementary result for the case of beams which are not slender.     

Large amplitudes free vibrations and post-buckling analysis of unsymmetrically laminated composite beams on nonlinear elastic foundation

The purpose of this paper is to present efficient and accurate analytical expressions for large amplitude free vibration and post-buckling analysis of unsymmetrically laminated composite beams on elastic foundation. Geometric nonlinearity is considered using Von Karman’s strain–displacement relations. Besides, the elastic foundation has cubic nonlinearity with shearing layer. The nonlinear governing equation is solved by employing the variational iteration method (VIM). This study shows that the third-order approximation of the VIM leads to highly accurate solutions which are valid for a wide range of vibration amplitudes. The effects of different parameters on the ratio of nonlinear to linear natural frequency of beams and the post-buckling load–deflection relation are studied.     

Nonlinear elliptic differential equations with multivalued nonlinearities

In this paper we study nonlinear elliptic boundary value problems with monotone and nonmonotone multivalued nonlinearities. First we consider the case of monotone nonlinearities. In the first result we assume that the multivalued nonlinearity is defined on all ℝ. Assuming the existence of an upper and of a lower solution, we prove the existence of a solution between them. Also for a special version of the problem, we prove the existence of extremal solutions in the order interval formed by the upper and lower solutions. Then we drop the requirement that the monotone nonlinearity is defined on all of ℝ. This case is important because it covers variational inequalities. Using the theory of operators of monotone type we show that the problem has a solution. Finally in the last part we consider an eigenvalue problem with a nonmonotone multivalued nonlinearity. Using the critical point theory for nonsmooth locally Lipschitz functionals we prove the existence of at least two nontrivial solutions (multiplicity theorem).     

Optimal Fine‐Scale Structures in Compliance Minimization for a Shear Load

We return to a classic problem of structural optimization whose solution requires microstructure. It is well‐known that perimeter penalization assures the existence of an optimal design. We are interested in the regime where the perimeter penalization is weak; i.e., in the effect of perimeter as a selection mechanism in structural optimization. To explore this topic in a simple yet challenging example, we focus on a two‐dimensional elastic shape optimization problem involving the optimal removal of material from a rectangular region loaded in shear. We consider the minimization of a weighted sum of volume, perimeter, and compliance (i.e., the work done by the load), focusing on the behavior as the weight ɛ of the perimeter term tends to 0. Our main result concerns the scaling of the optimal value with respect to ɛ. Our analysis combines an upper bound and a lower bound. The upper bound is proved by finding a near‐optimal structure, which resembles a rank‐2 laminate except that the approximate interfaces are replaced by branching constructions. The lower bound, which shows that no other microstructure can be much better, uses arguments based on the Hashin‐Shtrikman variational principle. The regime being considered here is particularly difficult to explore numerically due to the intrinsic nonconvexity of structural optimization and the spatial complexity of the optimal structures. While perimeter has been considered as a selection mechanism in other problems involving microstructure, the example considered here is novel because optimality seems to require the use of two well‐separated length scales.© 2016 Wiley Periodicals, Inc.     

Inflow problem for the one-dimensional compressible Navier–Stokes equations under large initial perturbation

This paper is concerned with the inflow problem for the one-dimensional compressible Navier–Stokes equations. For such a problem, Matsumura and Nishihara showed in [10] that there exists boundary layer solution to the inflow problem, and that both the boundary layer solution, the rarefaction wave, and the superposition of boundary layer solution and rarefaction wave are nonlinear stable under small initial perturbation. The main purpose of this paper is to show that similar stability results for the boundary layer solution and the supersonic rarefaction wave still hold for a class of large initial perturbation which can allow the initial density to have large oscillation. The proofs are given by an elementary energy method and the key point is to deduce the desired lower and upper bounds on the density function.     

Blow-up analysis for an elliptic equation describing stationary vortex flows with variable intensities in 2D-turbulence

We consider the mean field equation arising in the high-energy scaling limit of point vortices with a general circulation constraint, when the circulation number density is subject to a probability measure. Mathematically, such an equation is a non-local elliptic equation containing an exponential nonlinearity which depends on this probability measure. We analyze the behavior of blow-up sequences of solutions in relation to the circulation numbers. As an application of our analysis we derive an improved Trudinger-Moser inequality for the associated variational functional.     

Analytic proof of dual variational formula for the first eigenvalue in dimension one

The first non-zero eigenvalue is the leading term in the spectrum of a self-adjoint operator. It plays a critical role in various applications and is treated in a large number of textbooks. There is a well-known variational formula for it (called the Min-Max Principle) which is especially effective for an upper bound of the eigenvalue. However, for the lower bound of the spectral gap, some dual variational formulas have been obtained only very recently. The original proofs are probabilistic. Some analytic proofs in one-dimensional case are proposed and certain extension is made. Project supported in part by the National Natural Science Foundation of China (Grant No. 19631060), Qiu Shi Science & Technology Foundation, DPFIHE, MCSEC and MCMCAS.     

Optimal use of the translation method and relaxations of variational problems

The translation method has been used with great success in bounding the effective moduli of composite materials. We consider here the analogous method for bounding the relaxations of variational problems. We optimize the bound over the set of all available translations. Our method is to cast this in the form of a minmax problem. Using techniques of nonsmooth analysis, we are able to identify the optimal translation bound, meanwhile proving the existence of at least one optimal combination rank-one convex quadratic and null-Lagrangian translation. The optimal translation bound proves to be a better general lower bound on relaxations of variational problems than is the polyconvexification in three dimensions. In two dimensions, we discuss the negative result that the optimal translation bound is exactly the polyconvexification. Several examples of optimal applications of translation bounds to non-convex nonlinear variational problems are given.     

Blister Patterns and Energy Minimization in Compressed Thin Films on Compliant Substrates

This paper is motivated by the complex blister patterns sometimes seen in thin elastic films on thick, compliant substrates. These patterns are often induced by an elastic misfit that compresses the film. Blistering permits the film to expand locally, reducing the elastic energy of the system. It is therefore natural to ask: what is the minimum elastic energy achievable by blistering on a fixed area fraction of the substrate? This is a variational problem involving both the elastic deformation of the film and substrate and the geometry of the blistered region. It involves three small parameters: the nondimensionalized thickness of the film, the compliance ratio of the film/substrate pair, and the mismatch strain. In formulating the problem, we use a small‐slope (Föppl–von Kármán) approximation for the elastic energy of the film, and a local approximation for the elastic energy of the substrate. For a one‐dimensional version of the problem, we obtain “matching” upper and lower bounds on the minimum energy, in the sense that both bounds have the same scaling behavior with respect to the small parameters. The upper bound is straightforward and familiar: it is achieved by periodic blistering on a specific length scale. The lower bound is more subtle, since it must be proved without any assumption on the geometry of the blistered region. For a two‐dimensional version of the problem, our results are less complete. Our upper and lower bounds only “match” in their scaling with respect to the nondimensionalized thickness, not in the dependence on the compliance ratio and the mismatch strain. The lower bound is an easy consequence of our one‐dimensional analysis. The upper bound considers a two‐dimensional lattice of blisters and uses ideas from the literature on the folding or “crumpling” of a confined elastic sheet. Our main two‐dimensional result is that in a certain parameter regime, the elastic energy of this lattice is significantly lower than that of a few large blisters. © 2015 Wiley Periodicals, Inc.     

Application of homotopy-perturbation and variational iteration methods to nonlinear heat transfer and porous media equations

Perturbation methods depend on a small parameter which is difficult to be found for real-life nonlinear problems. To overcome this shortcoming, two new but powerful analytical methods are introduced to solve nonlinear heat transfer problems in this article; one is He's variational iteration method (VIM) and the other is the homotopy-perturbation method (HPM). The VIM is to construct correction functionals using general Lagrange multipliers identified optimally via the variational theory, and the initial approximations can be freely chosen with unknown constants. The HPM deforms a difficult problem into a simple problem which can be easily solved. Nonlinear convective–radiative cooling equation, nonlinear heat equation (porous media equation) and nonlinear heat equation with cubic nonlinearity are used as examples to illustrate the simple solution procedures. Comparison of the applied methods with exact solutions reveals that both methods are tremendously effective.     

The virial theorem and ground state energy estimates of nonlinear Schrödinger equations in $$\mathbb {R}^2$$ with square root and saturable nonlinearities in nonlinear optics

The virial theorem is a nice property for the linear Schrödinger equation in atomic and molecular physics as it gives an elegant ratio between the kinetic and potential energies and is useful in assessing the quality of numerically computed eigenvalues. If the governing equation is a nonlinear Schrödinger equation with power-law nonlinearity, then a similar ratio can be obtained but there seems to be no way of getting any eigenvalue estimates. It is surprising as far as we are concerned that when the nonlinearity is either square-root or saturable nonlinearity (not a power-law), one can develop a virial theorem and eigenvalue estimates of nonlinear Schrödinger (NLS) equations in \({{\mathbb {R}}^{2}}\) with square-root and saturable nonlinearity, respectively. Furthermore, we show here that the eigenvalue estimates can be used to obtain the 2nd order term (which is of order \(\ln \Gamma \)) of the lower bound of the ground state energy as the coefficient \(\Gamma \) of the nonlinear term tends to infinity.     

On the eigenvalue set structure for higher-order equations of elliptic type with discontinuous nonlinearities

We study the existence of nonzero solutions of the Dirichlet problem for a higherorder equation of elliptic type with a discontinuous nonlinearity. By the variational method, we prove the existence of a ray of positive eigenvalues and prove an upper bound for the bifurcation parameter in the problem.     

Analysis of large amplitude free vibrations of clamped tapered beams on a nonlinear elastic foundation

The purpose of this paper is to present efficient and accurate analytical expressions for large amplitude free vibration analysis of single and double tapered beams on elastic foundation. Geometric nonlinearity is considered using the condition of inextensibility of neutral axis. Moreover, the elastic foundation consists of a linear and cubic nonlinear parts together with a shearing layer. The nonlinear governing equation is solved by employing the variational iteration method (VIM). This study shows that the second-order approximation of the VIM leads to highly accurate solutions which are valid for a wide range of vibration amplitudes. The effects of different parameters on the nonlinear natural frequency of the beams are studied under different mode shapes. The results of the present work are also compared with those available in the literature and a good agreement is observed.     

A Lamination Upper Bound to the Free Energy of Shape Memory Alloys

Modeling the energetic behavior of materials showing martensitic phase transformations usually leads to non-convex energy formulations. In a variety of models based on quasi-convex analysis, the Reuß lower bound, which neglects the compatibility constraint for the deformation fluctuations, is used as an estimate for the so-called energy of mixing. We present an upper bound that is on the one hand based on the lamination mixture formula, which gives an estimate of the free energy of two-variant materials and is extended to a specialized n-variant case in our work. On the other hand, we rely on experimentally well established assumptions about the type of microstructure that forms in such alloys. More precisely, we restrict the set of physically admissible microstructures to the subset of second order laminated microstructres consisting of austenite and twinned martensites. We further refine our upper bound by taking into account the notion of twin-compatibility. For the physically relevant examples of 13- and 7-variant Cu-Al-Ni shape memory alloys, striking congruence is obtained in the comparison of the Reuß lower and our upper bound for fixed volume fractions. Furthermore, we show results of global minimization of the energy obtained by each bound over the volume fractions of the variants. Similarities and differences in the energy-minimizing volume fractions are discussed. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On a singular parabolic p-biharmonic equation with logarithmic nonlinearity

This paper is concerned with the well-posedness and asymptotic behavior of Dirichlet initial boundary value problem for a singular parabolic p-biharmonic equation with logarithmic nonlinearity. We establish the local solvability by the technique of cut-off combining with the methods of Faedo–Galerkin approximation and multiplier. Meantime, by virtue of the family of potential wells, we use the technique of modified differential inequality and improved logarithmic Sobolev inequality to obtain the global solvability, infinite and finite time blow-up phenomena, and derive the upper bound of blow-up time as well as the estimate of blow-up rate. Furthermore, the results of blow-up with arbitrary initial energy and extinction phenomena are presented.     

On the Two-Dimensional Navier—Stokes Equations with the Free Boundary Condition

In this article we consider the two-dimensional Navier—Stokes equations with free boundary condition (open surface), and derive a number of different results: a new orthogonal property for the nonlinear term, improved a priori estimates on the solution, an upper bound on the dimension of the attractor which agrees with the conventional theory of turbulence; finally, for elongated rectangular domains, an improved Lieb—Thirring (collective Sobolev) inequality leads to an upper bound on the dimension of the attractor which might be optimal. Accepted 11 July 1996     

A priori estimates and existence of positive solutions for strongly nonlinear problems

In this paper we study the existence of positive solutions for a nonlinear Dirichlet problem involving the m-Laplacian. The nonlinearity considered depends on the first derivatives; in such case, variational methods cannot be applied. So, we make use of topological methods to prove the existence of solutions. We combine a blow-up argument and a Liouville-type theorem to obtain a priori estimates. Some Harnack-type inequalities which are needed in our reasonings are also proved.