Optimal Dissection Algorithm Of Finite Element Calculative Mesh In Optimal Design

In recent years, we always use finite element methods to calculate stress and changes of location in the optimal design of construction. When using the finite element methods, we find that the amount of the information of element mesh node number and node coordinate is large, tedious and annoying, and it will make a lot of mistakes by manual work, so the calculative mesh automatic dissection is an important problem of the finite element calculation. For each iteration of optimal design accompanies reform of the calculative mesh, we must find a new more efficient automatic method of the calculative mesh dissection.There are two main kinds of automatic mesh dissection methods currently, and one is Functional Mapping Method, and the other is Nonfunctional Mapping Method. Both of them can solve the problems of automatic element establishment efficiently. Nevertheless, these methods still have some limitations, such as: most methods are unsuitable to multi-area structures; they are unable to control density of the nodes; the optimal judgment criterion is ambiguous.As above, in this article we bring a new method named Hold-Cover Method. We apply this method to find the optimal location coordinate of the mesh node by using the optimal control theory, and then we can obtain ideal distribution of nodes, furthermore this method can satisfy the demand of the multi-areas, multi-materials and the control of the node density causally.Using Hold-Cover Method needs a template mesh as the structure shape, and the coordinate of template mesh is set up in the assumed coordinate system automatically. In this article we bring the recursive formula of the template mesh element nodes number and nodes coordinate, which is set up automatically.When in actual calculation, we need to apply template mesh according to the actual condition of the construction, that means we select the part of the template mesh corresponding to the construction and choose the relativelines and borders of the calculative outlines as the control boundaries, and give out the node coordinate on the boundaries, and then transform the template mesh coordinate to the actual calculative coordinate system. Finally we adjust and optimize the node coordinate on the base above.When in the finite element calculation, we need the element shape approaches the regular triangle, furthermore the minimal the sum of the distance between nodes is, the most the element shape approaches the regular triangle. Therefore the research, the distribution method of the nodes with the minimal sum of distance between nodes, is the main problem in this article.1. Mathematical Model of the Nodes Distribution on Optimal ControlAssume that the researched area is closed, and it's nodes coordinates are consecutive in the calculative area. The adjustment of the nodes is by the hold of the neighbor nodes. The neighbor nodes around hold each node in the procession of the hold. Hold coefficient is bi jrs, which is affection of the node j coordinate on the direction s to the node i coordinate on the direction r , then the expression is below:For x and y are orthogonal, then bi jyx= 0, bi jyx= 0. According to all the neighbor nodes to node i , we can get the expression below:Expression(2)and expression(3)show that node i's coordinate depends on the adjustment of the neighbor nodes.The template mesh nodes coordinates have initial values after translation, and that shows we know the initial condition of the problem. When the sum of distance between node and neighbor nodes decreases, the changes of nodes coordinate decrease, and the margins of each node between the neighbor nodes decrease too, so the accumulative value of these margins algebraic sum will reach the maximum gradually. Obviously, this is a multivariate discrete system, and the problem is described as below:â‘´.Functional expression of the linear discrete systemThe problem can be concluded as blew:⑶.Terminal condition is free This problem is free terminal condition and also to research the coordinate of the optimal nodes.â‘·.Restriction x ( k ) needs to satisfy the demand of the control boundary nodes coordinates: X_r ( k )= X_r0 (6) r——Number of the control boundary nodes.⑸.Demand of AnswerWhen J [ x_i ] in function(4)reaches the maximum, x ( k )must satisfy necessary condition. According to variation method of multivariate discrete system and Euler equation of discrete system, terminal and initial conditions, we can get x ( k ). Function:According to Euler equation and terminal condition we can get:Replace k with k + 1: Expression(9)and expression(10)are the recursive formulas of the Hold-Cover Method to make finite element optimal dissection.2. Realization of the Hold and Control of the iteration The problem can be simplified, and after mesh is translated, nodes coordinates become initial condition x ( k₀ )= x0. Then according to recursive formula (9)and formula(10), we get x ( k₀ + 1), x ( k₀ + 2), , x ( k₁) gradually, which is shows the new adjusted node coordinate is the average of s neighbor nodes coordinate of last iteration. From the first node, we recalculate the coordinate to all the nodes as the expression (9) and (10), if some node is empty node or the boundary node, it will not be adjusted.Until all nodes satisfy the control criterion, we can obtain the optimal node location and element shape.The control object determines iteration control criterion. Selections of the accumulative value and component value are different as the object control error criterion, and we can use absolute criterion or relative criterion. Absolute criterion is the regulation of the actual change, while relative criterion is the ratio of change value to original value, and we often use relative criterion.According to the experience, we can confirm the control criterionεk. when iteration maximal errorεmax≤εk, the iteration must be terminated。3. Analyses and Sum-upIn this article, we use Hold-Cover Method to calculate the dissection of a gravity dam. After 50 iterations, the error of the iteration changes little. After 75 iterations, we obtain the relative error, which reaches the control criterion 0.0001, and absolute error 0.0011 meters. That outcome satisfies the demand of calculation.Hold-Cover Method uses variation method of the multivariable discrete system to solve the problem of the optimal dissection of the finite element calculative mesh. Actual application shows the effect of the dissection is ideal, and element shape approaches to the regular triangle, and element at the foot of downstream dam is small with dense nodes, and that satisfies the demand of finite element calculation. The function of the hold of nodes on the boundary to the inner nodes is obvious. Number of area material is correct, and area"0"is deleted already, and that satisfies the demand of calculation.Using Hold-Cover Method can form the calculative mesh rapidly, and it can be used as the front-deal tool in the large finite element calculative software at home and abroad, and especially it can promote the efficiency of calculation in optimal design.