Modeling and analysis of the unilateral contact of a piezoelectric body with a conductive support

We consider a mathematical model which describes the quasistatic process of contact between a piezoelectric body and an electrically conductive support, the so-called foundation. We model the material's behavior with a nonlinear electro-viscoelastic constitutive law; the contact is frictionless and is described with the Signorini condition and a regularized electrical conductivity condition. We derive a variational formulation for the problem and then we prove the existence of a unique weak solution to the model. The proof is based on arguments of nonlinear equations with multivalued maximal monotone operators and fixed point. Then we introduce a fully discrete scheme, based on the finite element method to approximate the spatial variable and the backward Euler scheme to discretize the time derivatives. We treat the unilateral contact conditions by using an augmented Lagrangian approach. We implement this scheme in a numerical code then we present numerical simulations in the study of two-dimensional test problems, together with various comments and interpretations.     

异步并行牛顿法的单调收敛性

本文对非线性方程组ＦＸ＝０提出异步并行牛顿法的单调型算法,算法的整体收敛性及局部超线性收敛性的证明。     

Non-linear traces on the algebra of compact operators and majorization

We study non-linear traces of the Choquet type and the Sugeno type on the algebra of compact operators. They have certain partial additivities. We show that these partial additivities characterize non-linear traces of both the Choquet type and the Sugeno type respectively. There exists a close relation between non-linear traces of the Choquet type and majorization theory. We study trace class operators for non-linear traces of the Choquet type. More generally we discuss Schatten–von Neumann $p$-class operators for non-linear traces of the Choquet type. We determine when they form Banach spaces. This is an attempt at non-commutative integration theory for non-linear traces of the Choquet type on the algebra of compact operators. We also consider the triangle inequality for non-linear traces of the Sugeno type.