Research On Composite Convex Programs And DC Programs As Well As Elliptical Inclusion In Magnetoelectroelastic Materials

This dissertation deals with composite convex optimization problems and DC (dif-ference of two convex functions) programs and establishes some optimality conditionsfor the strong duality and stable duality. Elliptical inclusion in magnetoelectroelastic ma-terials is also investigated, some analytical solutions are given and optimization resultsobtained are applied to show the existence of maximum energy release rate, which pro-vide references for the applied research of the civil, chemical, mechanical and aerospaceengineering, etc.For a DC problem, by using the properties of dualizing prarametrization functions,Lagrangian functions，the epigraph technique and characteristic sets, a dual problem isintroduced and some sufcient and necessary conditions (semi-closure condition, asymp-totic closure condition and closure condition) of the weak duality, stable zero duality gapand stable strong duality are also given. Some global constraint qualification in terms ofsubdiferential are given to ensure total duality.Since the composite functions of above problem are only linear composite function-s, we consider a convex composite programming problem with a cone-convex constraint.By using Fenchel conjugate transforms and epigraph of convex functions, a sufcient andnecessary condition of the stable Farkas lemma is given. This condition is weaker than theclassic Slater condition. Moreover, the condition obtained is also shown to be equivalentto the stable duality results. An example is given to show that the cone monotone increas-ing property of outer function is essential. Meanwhile, main results obtained developsome recently results.Based on the above two models of problems, we consider a convex composite D-C optimization problem with a cone-convex constraint. In the case when the functionsinvolved are lower semicontinuous, a closedness-type condition in terms of epigraph ofconjugate functions is introduced and proved to be sufcient and necessary for the strongduality between a convex composite optimization subproblem and its dual problem. How-ever this closedness-type condition is only sufcient to ensure the strong duality betweenthe original convex composite DC optimization problem and its dual problem. In the casewhen the lower semicontinuity of the functions involved and the cone monotone increas-ing outer function are not assumed, we first define the Fenchel-Lagrange duality of primal problem. Some sufcient and necessary conditions (semi-closure condition, further reg-ularity condition and closure condition) of weak duality, strong duality and stable strongduality results are derived by using epigraph and characteristic sets associated with theprimal problem and dual problem. Furthermore, Some subdiferential conditions are al-so given to ensure the stable total duality. The results obtained are applied to study themin-max optimization problem and l1penalty function problem.As optimization methods applications in mechanics, we consider the magnetoelec-troelastic elliptical cylinder inclusion in magnetoelectroelastic material, under the anti-plane loads. Analytic solutions in terms of analytic complex functions are obtained byusing conformal mapping and complex variable method. Inside the inclusion, the strain,electric and magnetic field are found to be uniform and vary with the shape of the ellipse.When the inclusion is reduced to a crack, along the interface, the strain, electric fieldstrength and magnetic field strength equal to the corresponding remote ones, which canbe used as a boundary condition. Special cases, such as an impermeable elliptical cavity,a rigid and permeable inclusion, a line inclusion and a crack problem are discussed indetail. For the general cases including the mode I, mode II or mode III, analytic solutionsfor matrix and inclusion are derived by using Stroh formalism and boundary conditions.Inside the inclusion, the strain, electric and magnetic field are found to be uniform andvary with the shape of the ellipse. All results are expressed in closed-form so that mate-rials scientists and engineers can easily use them for the design and tailoring of advancedmagnetoelectroelastic materials. Furthermore, crack problems are also investigated. Thestress, electric and magnetic fields in the vicinity of the crack tip are determined by acomplex vector of intensity factors. Various special cases, including a rigid and perme-able inclusion, line inclusion and soft permeable inclusion are discussed.At last optimization results obtained are applied to study the inclusion problem andverify the existence of the maximum energy release rate by taking anti-plane crack forexample.