Sequence Of Linear Subproblems Research Articles

Well-posedness result for the flow of Maxwell viscoelastic fluid of White-Metzner type

In this paper, we investigate the flow characteristics of an unsteady quasistatic Maxwell viscoelastic fluid of White-Metzner type within a three-dimensional bounded domain. Our objective is to establish the well-posedness of the resulting partial differential equations initial boundary value problem, both locally and globally in time, under the condition of small initial data. To demonstrate the existence of a solution, we employ an iterative method that transforms our problem into a sequence of linear subproblems. At each iteration, we tackle both elliptic and hyperbolic problems. The convergence of this iterative process provides the solution to the original problem.

An approximating pseudospectral method with state‐dependent coefficient optimization for nonlinear optimal control problem

AbstractThe approximating sequence Riccati equation method is an efficient approach for solving the nonlinear optimal control problems, but its neglect of nonlinear dynamics and necessary optimality condition makes the control law difficult to satisfy the optimality. In this paper, an approximating pseudospectral method with state‐dependent coefficient optimization algorithm is proposed to solve this defect. By introducing the approximating pseudospectral method, the original nonlinear problem is transformed into a sequence of linear subproblems, which preserves the nonlinearity of solution. Then a state‐dependent coefficient optimization algorithm based on the gradient projection technique is proposed, which ensures the optimality of the control law. A double‐layer optimization structure is designed to facilitate the coordination between the approximating method and the optimization algorithm. Theoretical analysis proves the convergence of the proposed method. Comparative case studies illustrate the effectiveness in reducing the performance index and ensuring the optimality of the control law.

A computational study of interfacial debonding damage in fibrous composite materials

In this paper the effect of finite interface strength, and possible debonding, on the macroscopic response of a sample of fiber-reinforced composite material is computationally investigated via the finite element method. The sample consists of several fibers embedded in a homogeneous matrix, aligned in the longitudinal direction, and randomly distributed in the transverse direction. Plane strain conditions are enforced. Both the matrix and the fibers are assumed to behave in a linearly elastic manner. The approach is to employ unilateral constraints to model interface strength limits. However, because the debonded surfaces are unknown a priori, and depend on the internal fields, the originally linear elastic problem becomes nonlinear, and hence it must be solved in an iterative manner. Accordingly, a nested contact algorithm scheme is developed, based on an active set strategy, to efficiently simulate multiple interacting unilateral constraints. The nesting allows the nonlinear problem within a Newton step to be transformed into a sequence of linear sub-problems. Using the algorithm, numerical tests are performed on a widely used Aluminum/Boron fiber-reinforced composite combination to determine the effects of debonding on changes in macroscopic responses as a function of interface strength and loading. It is shown that the amount of debonded surface area correlates perfectly with the loss in the macroscopic stiffness of the material. This result lends credence to damage evolution laws, for homogenized material models, which employ interface separation surface area as the primary internal damage variable.