Mathematical Problems in Engineering

Volume 2017 (2017), Article ID 4829195, 10 pages

https://doi.org/10.1155/2017/4829195

## An Improved Finite Element Meshing Strategy for Dynamic Optimization Problems

School of Automation, Hangzhou Dianzi University, Hangzhou 310018, China

Correspondence should be addressed to Aipeng Jiang

Received 7 April 2017; Accepted 11 June 2017; Published 19 July 2017

Academic Editor: Rahmat Ellahi

Copyright © 2017 Minliang Gong et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The finite element orthogonal collocation method is widely used in the discretization of differential algebraic equations (DAEs), while the discrete strategy significantly affects the accuracy and efficiency of the results. In this work, a finite element meshing method with error estimation on noncollocation point is proposed and several cases were studied. Firstly, the simultaneous strategy based on the finite element is used to transform the differential and algebraic optimization problems (DAOPs) into large scale nonlinear programming problems. Then, the state variables of the reaction process are obtained by simulating with fixed control variables. The noncollocation points are introduced to compute the error estimates of the state variables at noncollocation points. Finally, in order to improve the computational accuracy with less finite element, moving finite element strategy was used for dynamically adjusting the length of finite element appropriately to satisfy the set margin of error. The proposed strategy is applied to two classical control problems and a large scale reverse osmosis seawater desalination process. Computing result shows that the proposed strategy can effectively reduce the computing effort with satisfied accuracy for dynamic optimization problems.

#### 1. Introduction

The direct transcription method is an important method to solve the problem of optimal control. By discretization of differential and algebraic optimization problems (DAOPs), the state variables and control variables are completely discretized. The discrete method uses the finite element orthogonal collocation, and generally the number of finite elements is empirically selected and the length of each finite element is equally divided. This results in low discretization accuracy for state and control variables, and to guarantee the satisfactory accuracy for some problems, the calculation time is too long to accept. Moving finite element strategy is a good idea for the solution. With the need of discrete differential algebraic equations, moving finite element is becoming the popular and practical technique for chemical process.

At present, we are concerned with calculation accuracy not only of the material, but also of the time in the process parameters for the chemical process attention. Modeling methods based on first principle and data-driven are used for practical control. With the development of solving technology, we can better understand the changes of state variables in the whole process of chemical reaction.

Betts and Kolmanovsky proposed a refinement procedure for nonlinear programming for discrete processes and estimating the discretization error for state variables [1]. Liu et al. used a novel penalty method to deal with nonlinear dynamic optimization problems with inequality path constraints [2]. Paiva and Fontes studied the adaptive mesh refinement algorithms which allow a nonuniform node collocation and apply a time mesh refinement strategy based on the local error into practical problems [3]. Zhao and Tsiotras introduced an efficient and simple method based on density (or monitor) functions, which have been used extensively for grid refinement. The accuracy and stability of the method were improved by choosing the appropriate density function. [4]. When using the discrete method to solve the nonlinear problem, iterative programming (IDP) algorithm is rather vulnerable to the stage of time in several aspects such as accuracy and the convergence rate. Li et al. introduce a self-adaptive variable-step IDP algorithm, taking account of the performance and control variables [5]. Zhang et al. presented an adaptive variable-step-size RKF method based on norm control. The method automatically adjusts the step size and the case where the local error norm value is 0 or the minimum value is discussed [6]. In view of the optimization of biochemical processes, some researchers have proposed adaptive parameterization methods to solve such problems [7, 8].

The above work is quite helpful to quick and stable solution of the dynamic optimization problem, but most of them put emphasis on method of choice and do not propose a specific operational process. As we know, the results of the optimization problem are often dependent on the specific algorithm.

This paper is a mesh refinement strategy based on the variable finite element mesh method [9–11]. Prior to optimization, a suitable initial mesh and length are obtained by GAMS platform simulation. Then, to optimize the objective function, the finite element is moved appropriately under the condition that the set error is satisfied. The method is applied to the chemical reaction of simple ODE equation [12, 13] and large scale reverse osmosis seawater desalination process, which proves its validity and feasibility.

#### 2. Finite Element Discrete Method for Equation Model

According to the whole chemical reaction process, we want to know the changes in the material. Discrete solution of the model is the key part for computing process. Here, we introduce the orthogonal polynomials based on Lagrange’s use of the Radau collocation points on the finite element [14] and use of simultaneous method to the entire time domain . The simplified model can be written as the following equations:where is the scalar objective function, and as differential and algebraic variables, respectively; is the constraint equation for the state variables and the control variables, and as a control variable and time independent optimization variable. is a polynomial of order in collocation . denotes the length of element . is value of its first derivative in element at the collocation point . The advantage of the Lagrange interpolation polynomial over other interpolation methods is that the value of the variable at each collocation point is exactly equal to its coefficient

#### 3. Error Calculation at Noncollocation Point

On the selection of collocation points on finite elements, if the mathematical expression is configured according to the Gauss point, the algebraic precision of the numerical integration is the highest. Gauss points can achieve the algebraic precision of order. When using Radau points to configure, the algebraic precision is lower than the Gauss point, . The discretization proposition based on Radau point has better stability [15]. Therefore, this paper uses Radau points to configure.

In order to improve the accuracy of the solution, Vasantharajan and Biegler [16] inserted the noncollocation points on the finite element to determine the increase and decrease of the finite element by the error of the noncollocation point.

Generally, in the solution of problem (1), it will cause compute error on state or control profiles due to finite mesh. In this study, the error of collocation finite elements can be estimated fromHere is constant depending on collocation points. is differential state equation at

From (3), will be zero at the collocations. So we need to select a noncollocation point ; that is, . Specifically, can be stated aswhere is the differential of . Then we add (3)-(4) to problem (1) and then resolve the OPC problem.

#### 4. Finite Element Method

As the error estimation gotten with (3) and (4), we can divide the finite elements mesh. Generally, equally spaced method was widely used in many research, due to its simply operation. The flow chart of calculation is shown in Figure 1.