Since the introduction of the finite element approach, as a numerical solution scheme for structural and solid mechanics applications, various for mulation methodologies have been proposed. These ways offer different advantages and shortcomings. Among these techniques, the standard displacement-based approach has attracted more interest due to its straightforward scheme and generality. Investigators have proved that the other strategies, such as the force-based, hybrid, assumed stress, and as sumed strain provides special advantages in comparison with the classicfinite elements. For instance, the mentioned techniques are able to solve difficulties, like shear locking, shear parasitic error, mesh sensitivity, poor convergence, and rotational dependency. The main goal of this two-part study is to present a brief yet clear portrait of the basics and advantages of the direct strain-based method for development of high-performance plane finite elements. In this article, which is the first part of this study, assump tions and the basics of this method are introduced. Then, a detailed review of all the existing strain-based membrane elements is presented. Although the strain formulation is applicable for different types of structures, most of the existing elements pertain to the plane structures. The second part of this study deals with the application and performance of the reviewed elements in the analysis of plane stress/strain problems.

This study presents an algorithm for solving optimal control problems with the objective function of the Lagrange-type and multiple delays on both the state and control variables of the constraints, with bounds on the control variable. The full discretization of the objective functional and the multiple delay constraints is carried out by using the Simpson numerical scheme. The discrete recurrence relations generated from the discretization of both the objective functional and constraints are used to develop the matrix operators, which satisfy the basic spectral properties. The primal-dual residuals of the algorithm are derived in order to ascertain the rate of convergence of the algorithm, which performs faster when relaxed with an accelerator variant in the sense of Nesterov. The direct numerical approach for handling the multi-delay control problem is observed to obtain an accurate result at a faster rate of convergence when over-relaxed with an accelerator variant. This research problem is limited to linear constraints and objective functional of the Lagrange-type and can address real-life models with multiple delays as applicable to quadratic optimization of intensity modulated radiation theory planning. The novelty of this research paper lies in the method of discretization and its adaptation to handle linearly and proximal bound-constrained program formulated from the multiple delay optimal control problems.