The FPES (fast point and element search) method is a useful and efficient strategy for node field transfer from old mesh to the new mesh in adaptive remeshing procedure. The FE mesh after adaptive remeshing with various error estimates will be refined at local region, and the mesh after adaptive remeshing has the heterogeneous density distribution. The FPES has the capacity to define the nearest search path adapting to the mesh with heterogeneous density distribution. It is a point location process which includes point searching, point location in element, and weight factor distribution. This strategy has been integrated to our finite element adaptive remeshing simulations, and it works well and rapidly. The three-dimension finite element numerical simulation of simply tensile test, orthogonal cutting, and metal milling process is given out to study its accuracy and efficiency.

Many physical phenomena in science and engineering can be modeled by partial differential equations and solved by means of the finite element method. In

Generally, state variables are stored at both nodal and integration points of element. In [

The querying efficiency is the major problem to be considered in FPES algorithm, especially, in 3D topology spatial with heterogeneous mesh density. The proposed FPES algorithm has the ability of defining the search path adapt to the spatial mesh density. It uses vector calculation to judge the spatial position of the nodes. After locating, distance-based weight factor replace volume calculation to compute the field value on nodes. These operations improve the transfer efficiency greatly and ensure the transfer accuracy strictly. In this paper, three main processes of FPES algorithm are presentd which include point-searching, point location in element and weight factor distributing. For testing, the proposed fast point and element search algorithm is integrated into our 3D finite element adaptive remeshing procedure. The numerical applications of simply tensile test, orthogonal cutting and metal milling process are given out to study its accuracy and efficiency. The interpolation method is also proposed to transfer nodal field for comparing the transfer accuracy with FPES method.

FPES (fast point and element search) method executes node-to-node field transfer accurately and quickly. We consider iteration

Some classical searching algorithms, such as counter clock wise [

There are two process in point search algorithm: preprocessing and querying process. Preprocessing read the mesh space and create adaptive container, include:

define adaptive orientation (

store all nodes label and its

define container capacity (

based on

Point querying process will locate

locate

based on

searching

Some point querying tests have been done to test the querying efficiency. Testing in the environment of 2.5 GHz CPU and 1.98 GB memory for example, when we query one point in the mesh space with 21354 nodes and 109371 elements, 462 ms need for preprocessing and 0.284 ms (it is the average value when we search 10000 points in 2840 ms) need for point querying process. To querying one point, it is obviously that the preprocessing occupy almost all of the searching time. But when millions of points querying, the preprocessing time can be neglected. From this point, it is very suitable for large-scale finite element analysis coupled with adaptive remeshing procedure.

This is the first step of FPES algorithm and some advanced work have been done to research the relationship between querying efficiency and different distributions of mesh density. Figure

Time searching in different mesh spaces with the increasing number of nodes.

Preprocessing

Point searching process

In our adaptive remeshing procedure, the mesh density is controlled by two error estimations, especially the prior estimation. After changing the prior estimation parameter, different mesh spaces with the increasing number of nodes and changed distribution of mesh density are generated to study the relationship between querying efficiency and different distributions of mesh density. Firstly, we define the container factor

Point searching in various mesh space.

Searching with various container factors

Optimized searching container factors

Some points of new mesh are located in the points of old mesh and others will be located in elements of old mesh. The general method of judging the relation of spatial location between point and tetrahedron is volumetric discriminance. Two kinds of volume must be calculated with this method, which include: the volume of tetrahedron; the sub-tetrahedron which formed by the point and surface of tetrahedron. When volumetric summation of sub tetrahedrons equals to the tetrahedron volume, the point will be judged in tetrahedron. For large-scale calculating, this method is obviously limited by its calculating efficiency. In this paper, the vectorial calculation is proposed to judge the point in element.

In our mesh space, the tetrahedral elements are constructed by four nodes which have the given orders. Four regular surfaces are defined to make the normal of surfaces point at the the same direction (out or in element). As in Figure

The judgement of the point in element.

Tetrahedral element

Surface and searching point

Based on (

In brief, the algorithm locates point

All of the nodes

locate at node

locate at edge of element

locate at surface of element

locate in element

We note

Finally, the state variable

In this paper, we are concentrated in searching and locating the points at the

The adaptive remeshing procedure integrates ABAQUS/Explicit solver, OPTIFORM adaptive mesher and field transformations, as shown in Figure

3D finite element simulation coupled with adaptive remeshing procedure.

In this paper, FPES algorithm is focused on transferring the temperature filed from old mesh to the new one. It is a dependent code which just relates to the spatial position of the nodes. Element killing and new boundary reconstructing in adaptive remeshing procedure have no effects on FPES. By contrary, the state variables, like temperature, will be removed following with the killed elements. This makes FPES become a useful tool of field transfer for simulating the metal removing process.

In this section, the transfer accuracy of FPES algorithm is discussed through a simply tensile test. Some comparatione of transfer accuracy are presented using orthogonal cutting test. At last, the transfer efficiency is described through metal milling test.

A simply tensile test using adaptive remeshing procedure is given out to research the transfer accuracy of FPES method. Initially, all of the nodes give the temperature of 200°C. As in Figure

Temperature transfer for simply tensile test using FPES code.

Comparing the temperature distributions before and after temperature transfer using FPES code in iteration 2, the temperature on 12 nodes in Figure

The interpolation method is also proposed to compare the transfer accuracy with FPES method. Three steps are included in node filed transfer using interpolation function. Firstly, shape function (as in equation) is used to transfer node field from nodes to interpolation points of element in the mesh before adaptive remeshing. Secondly, as one field of elements, the temperature are introduced into adaptive remeshing process and transferred to the interpolation points of new mesh. At last, the temperature on nodes are calculated in two parts: searching all of the elements which share this node; averaging the temperature on interpolation points of these elements. The shape function used is shown in (

The adaptive remeshing procedure for metal orthogonal cutting are presented to compare the transfer accuracy between FPES method and interpolation method. Initial, the workpiece has the very coarse mesh of 736 deformed tetrahedral elements. The cutting tool has 636 rigid hexahedral elements and only temperature degree is available. During the remeshing iterations, the mesh size of workpiece is controlled by geometrical and physical error estimations through the adaptive discrete parameters:

Iteration 10: temperature transfer for orthogonal cutting using both FPES method and interpolation method.

Before transfer

After transfer of FPES

Before transfer

After transfer of interpolation

Adaptive remeshing procedure and temperature transfer in orthogonal cutting: left, and metal up milling: right.

Some other iterations, shown in Table

Temperature lost using both FPES and interpolation methods.

Element kill | |||||||
---|---|---|---|---|---|---|---|

1 | 41.97 | 41.97 | 42.23 | 29.56 | No | ||

10 | 176.51 | 176.51 | 148.68 | 86.02 | No | ||

50 | 806.75 | 806.75 | 413.20 | 184.45 | Yes | ||

100 | 1220.76 | 1220.76 | 760.36 | 197.07 | Yes | ||

150 | 519.12 | 519.12 | 512.12 | 183.96 | Yes | ||

200 | 502.83 | 484.10 | 884.36 | 215.11 | Yes |

The up (conventional) milling adaptive remeshing procedure is presented to study the transfer efficiency of FPES algorithm. Initial, the workpiece has the very coarse mesh of 276 deformed tetrahedral elements. The milling tool has 2160 rigid hexahedral elements and only temperature degree is available. The finite meshes are refined and coarsen automatically in adaptive remeshing procedure. For this model, the adaptive discrete parameters are configured as:

In Table

Efficiency of node field transfer using FPES in different iterations.

50 | 7946 | 37549 | 172 | 6697 | 0.999 | 4 | 79 | 1166 | 9.396 |

100 | 9017 | 39536 | 180 | 7286 | 1.142 | 7 | 139 | 1585 | 10.941 |

150 | 11644 | 51065 | 208 | 10143 | 1.540 | 3 | 149 | 1349 | 11.352 |

200 | 14456 | 62221 | 232 | 12340 | 2.019 | 7 | 174 | 1935 | 15.847 |

The FPES (fast point and element search) method is a useful and efficient strategy to node field transfer from old mesh to the new mesh in adaptive remeshing procedure. The algorithm includes point-searching, point location in element and weight factor distributing. The nearest search path which adapts to the heterogeneous density distribution of mesh is defined. And the vector calculation and distance-based weight factor are used to improve querying efficiency and accuracy. The three-dimension finite element simulation of simply tensile test, orthogonal cutting and metal milling process is illustrated to discuss the transfer efficiency and accuracy.