CN112100889A - Optimization method of two-dimensional triangular mesh finite element model - Google Patents
Optimization method of two-dimensional triangular mesh finite element model Download PDFInfo
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Abstract
The invention discloses an optimization method of a two-dimensional triangular mesh finite element model, which comprises the following steps: 1) and acquiring an initial triangular mesh finite element model. 2) Establishing a node optimization model; 3) determining the normal degree of each node; 4) calculating degrees of all nodes; 5) computing node optimization model objective function
If it is
No reduction is carried out, step 8) is carried out, otherwise step 6) is carried out; 6) for the node v of the finite element model of the current triangular meshiOptimizing; 7) judging whether i is greater than or equal to n; if so, updating the triangular mesh finite element model, returning to the step 5), otherwise, making i equal to i +1, and returning to the step 6); 8) judging whether a non-acute-angle triangle exists, if so, turning to step 9), and otherwise, outputting the current triangular mesh finite element model; 9) optimizing the angle of the triangular grid unit and returning to the step8). The invention improves the overall harmony of the model, reduces the distortion degree of the model and can meet the actual industrial requirements.
Description
Technical Field
The invention relates to the field of finite element models, in particular to an optimization method of a two-dimensional triangular mesh finite element model.
Background
At present, the optimization method for a two-dimensional triangular mesh finite element model mainly changes the positions of nodes, and the connection mode of triangular units and the shapes of the triangular units are changed due to the change of the positions of the nodes, so the optimization method for changing the positions of the nodes is complex in algorithm; partial optimization methods start with local optimization, and for a given arbitrary initial model, poor local minimum points are easily trapped, resulting in a low quality of the generated mesh.
Therefore, there is a need for an optimization method that is simple and easy to implement and generates a high quality mesh.
Disclosure of Invention
The invention aims to provide an optimization method of a two-dimensional triangular mesh finite element model, which comprises the following steps:
1) and acquiring an initial triangular mesh finite element model.
2) And establishing a node optimization model.
The objective function Δ φ of the node optimization model is as follows:
in the formula (f)iIs a node viDegree of (c). The node degree represents the number of adjacent triangle elements with the node as one of the vertices. FiIs a node viThe normal degree of (c). And delta phi is the irregularity degree of the current triangular mesh finite element model.
3) Determining a normal degree of each node, and writing a normal degree set F ═ F1，F2，…，FnIn (c) }.
The steps of determining the degree of normality of each node are as follows:
3.1) judging node viType (c) of the cell.
When node viWhen it is an internal node, it is recordedGauge number FiAnd go to step 3) 6. When node viAnd if the node is a boundary node, the step 2) is carried out.
The internal nodes are nodes positioned in the initial triangular mesh finite element model. And the boundary nodes are nodes positioned on the boundary of the finite element model of the initial triangular mesh. i has an initial value of 1.
3.2) obtaining node viAngle between adjacent 2 boundary lines
When in use
Time, normal degree Fi＝2。
When in use
Time, normal degree Fi＝3。
When in use
Time, normal degree Fi＝4。
When in use
Time, normal degree Fi＝5。
When in use
Time, normal degree Fi＝6。
When in use
Time, normal degree Fi＝7。
At a certain node viAfter the normal degree, the process proceeds to step 3).
3.3) judging whether i is larger than or equal to n, if so, ending, otherwise, making i equal to i +1, and returning to the step 3.1).
4) Calculating degrees of all nodes of the current triangular mesh finite element model, and writing a degree set f ═ f into the degree set1，f2，…，fnIn (c) }. And n is the number of nodes of the current triangular mesh finite element model.
5) Inputting the degree set F and the regular degree set F of the initial triangular mesh finite element model into the node optimization model, and calculating the objective function of the node optimization model
If the objective function
No reduction is performed, step 8) is performed, otherwise step 6) is performed.
6) For node v of current triangular mesh finite element modeliAnd (6) optimizing. i has an initial value of 1.
The steps for optimizing the nodes of the current triangular mesh finite element model are as follows:
6.1) determining internal node viDegree of (f)iIf yes, ending the optimization; otherwise, turning to step 6.2);
6.2) determining internal node viDegree of (f)iIf the result is 3, if so, the step is shifted to a step 3), and if not, the step is shifted to a step 6.4);
6.3) delete internal node viAnd will be at internal node viThree triangular units as vertexes are fused into one triangular unit, and the optimization is finished;
6.4) determining internal node viDegree of (f)iIf yes, go to step 5), otherwise go to step 6.6);
6.5) delete internal node viAnd will be at internal node viFour triangular units as vertexes are fused into a quadrilateral unit; connecting a diagonal line of the quadrilateral unit, dividing the quadrilateral unit into 2 acute-angle triangular units, and finishing optimization;
6.6) determining node vtDegree of (f)tDegree of harmonyNumber Ft；
If number f of degreest＝Ft-1, respectively obtaining distance nodes vtNearest and satisfying degree fj1＝Fj1Node v of-1j1Distance internal node viNearest and satisfying degree fn1＝Fn1Node v of +1n1And distance node vj1Nearest and satisfying degree fm1＝Fm1Node v of +1m1(ii) a Connecting internal nodes vtNode vj1Node vn1Node vm1Forming a diagonal quadrilateral structure; wherein, the node vtAnd node vj1Are positioned on the same diagonal; node vn1And node vm1Are positioned on the same diagonal; node vtNode vn1Length of connecting line is less than or equal to node vtNode vm1The length of the connecting wire;
if number f of degreest＝Ft+1, respectively acquiring distance nodes vtNearest and satisfying degree fj2＝Fj2Node v of +1j2Distance node vtNearest and satisfying degree fn2＝Fn2Node v of-1n2And distance node vj2Nearest and satisfying degree fm2＝Fm2Node v of-1m2(ii) a Connecting node vtNode vj2Node vn2Node vm2Forming a diagonal quadrilateral structure; wherein, the node vtAnd node vj2Are positioned on the same diagonal; node vn2And node vm2Are positioned on the same diagonal; node vtNode vn2Length of connecting line is less than or equal to node vtNode vm2The length of the connecting wire;
6.7) determining a diagonal quadrilateral area, which comprises the following main steps:
6.7.1) selects to be located at the node vtAnd node vmxAnd node vt+q1Nodes which are connected and have the same degree as the normal degree are marked as vt+q1+1(ii) a q1 initial value is 0; x is 1 or 2;
6.7.2) judging q1>Whether Q1 holds, if so, thenStep 6.7.3) is entered, otherwise, q1 is made q1+1, and step 6.7.1) is returned; wherein, the node vt+Q1To be located at node vtAnd node vmxAnd node vmxNodes which are connected and have the same degree as the normal degree;
6.7.3) selects to be located at the node vnAnd node vjxAnd node vn+q2And node vt+q1Nodes which are connected and have the same degree as the normal degree are marked as vn+q2+1(ii) a q2 initial value is 0; x is 1 or 2; q1 and q2 are equal in value.
6.7.4) judging q2>If Q2 is true, if yes, go to step 6.7.5), otherwise, make Q2 become Q2+1, and return to step 6.7.3); wherein, the node vn+Q2To be located at node vnAnd node vjxAnd node vjxNodes which are connected and have the same degree as the normal degree;
6.7.5) to be controlled by node vt+q1Node vn+q2Node vjxNode vmxForming a diagonal quadrilateral area by the triangular units which are the vertexes; wherein Q1 is [0, 1, 2, …, Q1]；q2＝[0,1,2，…，Q2]；
6.8) dividing the diagonal quadrilateral area into quadrilateral units. Each quadrilateral unit is composed of two adjacent triangular units. Wherein, the diagonal line of the quadrangle is the common edge of two adjacent triangular units.
6.9) deleting the current diagonal line of each quadrilateral unit, connecting the other diagonal line of each quadrilateral unit, and finishing the optimization.
7) And judging whether i is greater than or equal to n. If yes, updating the triangular mesh finite element model and returning to the step 5), otherwise, making i equal to i +1 and returning to the step 6).
8) And judging whether the current triangular mesh finite element model has a non-acute angle triangle, if so, turning to the step 9), and otherwise, outputting the current triangular mesh finite element model.
9) The angles of the triangular mesh cells are optimized and the process returns to step 8).
The steps for optimizing the angle of the triangular grid unit are as follows:
9.1) establishing a plurality of triangular mesh unit clusters based on the current triangular mesh finite element model. Any one triangular mesh unit cluster is a convex polygon formed by three adjacent triangular mesh units. At least one of the three triangular grid units is a non-acute-angle triangle.
And 9.2) respectively using the middle point of each edge of the triangular grid unit cluster as the center of a circle and using the length of the edge where the center of the circle is located as the diameter to make a circle.
9.3) inserting nodes G sampled randomly in effective positions in the non-intersected areas of all circles by utilizing a gap processing method. And connecting the node G with the vertex of the triangular mesh unit set to complete the angle optimization of the triangular mesh unit.
9.4) repeating the step 9.2) and the step 9.3) until all the triangular mesh unit clusters are processed.
The technical effects of the invention are undoubted, and the invention respectively provides a node degree optimization method and a unit angle optimization method for inserting new nodes aiming at the problems that the generated initial triangular mesh finite element model has node degree irregularity and angle irregularity. The optimized model node degrees are regular degrees, and non-acute-angle triangles do not exist in the units, so that the overall harmony of the model is improved, the distortion degree of the model is reduced, and the actual industrial requirements can be met.
Drawings
FIG. 1 is a process flow diagram;
FIG. 2 is a diagram illustrating the values of the normal degrees of the boundary nodes;
FIG. 3(a) is a flow chart of cell fusion and subdivision optimization I;
FIG. 3(b) is a flow chart for unit fusion and subdivision optimization II;
FIG. 3(c) is a flow chart III for cell fusion and subdivision optimization;
FIG. 3(d) is a flow chart IV for cell fusion and subdivision optimization;
FIG. 4(a) is a finite element model before edge commutation.
FIG. 4(b) is a finite element model after edge inversion.
FIG. 5(a) is a finite element model before a new node is inserted.
FIG. 5(b) is a finite element model after the insertion of a new node.
FIG. 6(a) is a finite element model of an initial triangular mesh for a sample.
FIG. 6(b) is a triangular mesh finite element model after a certain sample is optimized.
Detailed Description
The present invention is further illustrated by the following examples, but it should not be construed that the scope of the above-described subject matter is limited to the following examples. Various substitutions and alterations can be made without departing from the technical idea of the invention and the scope of the invention is covered by the present invention according to the common technical knowledge and the conventional means in the field.
Example 1:
referring to fig. 1 to 6, a method for optimizing a two-dimensional triangular mesh finite element model includes the following steps:
1) and acquiring an initial triangular mesh finite element model of the workpiece.
2) And establishing a node optimization model.
The objective function Δ φ of the node optimization model is as follows:
in the formula (f)iIs a node viDegree of (c). The node degree represents the number of adjacent triangle elements with the node as one of the vertices. For example, fiRepresenting a node viQuilt fiCommon to adjacent delta units. FiIs a node viThe normal degree of (c). And delta phi is the irregularity degree of the current triangular mesh finite element model.
3) Determining a normal degree of each node, and writing a normal degree set F ═ F1，F2，…，FnIn (c) }.
The steps of determining the degree of normality of each node are as follows:
3.1) judging node viType (c) of the cell.
When node viWhen the node is an internal node, the normal degree F is recordediAnd go to step 3) 6. When node viAnd if the node is a boundary node, the step 2) is carried out.
The internal nodes are nodes positioned in the initial triangular mesh finite element model. And the boundary nodes are nodes positioned on the boundary of the finite element model of the initial triangular mesh. i has an initial value of 1.
3.2) obtaining node viAngle between adjacent 2 boundary lines
When in use
Time, normal degree Fi＝2。
When in use
Time, normal degree Fi＝3。
When in use
Time, normal degree Fi＝4。
When in use
Time, normal degree Fi＝5。
When in use
Time, normal degree Fi＝6。
When in use
Time, normal degree Fi＝7。
At a certain node viAfter the normal degree, the process proceeds to step 3).
3.3) judging whether i is larger than or equal to n, if so, ending, otherwise, making i equal to i +1, and returning to the step 3.1).
4) Calculating degrees of all nodes of the current triangular mesh finite element model, and writing a degree set f ═ f into the degree set1，f2，…，fnIn (c) }. And n is the number of nodes of the current triangular mesh finite element model.
5) Inputting the degree set F and the regular degree set F of the initial triangular mesh finite element model into the node optimization model, and calculating the objective function of the node optimization model
If the objective function
No reduction is performed, step 8) is performed, otherwise step 6) is performed.
6) For node v of current triangular mesh finite element modeliAnd (6) optimizing. i has an initial value of 1.
The steps for optimizing the nodes of the current triangular mesh finite element model are as follows:
the steps for optimizing the nodes of the current triangular mesh finite element model are as follows:
6.1) determining internal node viDegree of (f)iIf yes, ending the optimization; otherwise, turning to step 6.2);
6.2) determining internal node viDegree of (f)iIf the result is 3, if so, the step is shifted to a step 3), and if not, the step is shifted to a step 6.4);
6.3) delete internal node viAnd will be at internal node viThree triangular units as vertexes are fused into one triangular unit, and the optimization is finished;
6.4) determining internal node viDegree of (f)iIf yes, go to step 5), otherwise go to step 6.6);
6.5) delete internal node viAnd will be at internal node viFour triangular units as vertexes are fused into a quadrilateral unit; connecting a diagonal of the quadrilateral unit to divide the quadrilateral unit into2 acute angle triangular units are adopted, and optimization is finished;
6.6) determining node vtDegree of (f)tAnd normal degree Ft；
If number f of degreest＝Ft-1, respectively obtaining distance nodes vtNearest and satisfying degree fj1＝Fj1Node v of-1j1Distance internal node viNearest and satisfying degree fn1＝Fn1Node v of +1n1And distance node vj1Nearest and satisfying degree fm1＝Fm1Node v of +1m1(ii) a Connecting internal nodes vtNode vj1Node vn1Node vm1Forming a diagonal quadrilateral structure; wherein, the node vtAnd node vj1Are positioned on the same diagonal; node vn1And node vm1Are positioned on the same diagonal; node vtNode vn1Length of connecting line is less than or equal to node vtNode vm1The length of the connecting wire;
if number f of degreest＝Ft+1, respectively acquiring distance nodes vtNearest and satisfying degree fj2＝Fj2Node v of +1j2Distance node vtNearest and satisfying degree fn2＝Fn2Node v of-1n2And distance node vj2Nearest and satisfying degree fm2＝Fm2Node v of-1m2(ii) a Connecting node vtNode vj2Node vn2Node vm2Forming a diagonal quadrilateral structure; wherein, the node vtAnd node vj2Are positioned on the same diagonal; node vn2And node vm2Are positioned on the same diagonal; node vtNode vn2Length of connecting line is less than or equal to node vtNode vm2The length of the connecting wire;
6.7) determining a diagonal quadrilateral area, which comprises the following main steps:
6.7.1) selects to be located at the node vtAnd node vmxAnd node vt+q1Connected section with degree equal to normal degreePoints, denoted as vt+q1+1(ii) a q1 initial value is 0; x is 1 or 2;
6.7.2) judging q1>If Q1 is true, if yes, go to step 6.7.3), otherwise, make Q1 become Q1+1, and return to step 6.7.1); wherein, the node vt+Q1To be located at node vtAnd node vmxAnd node vmxNodes which are connected and have the same degree as the normal degree;
6.7.3) selects to be located at the node vnAnd node vjxAnd node vn+q2And node vt+q1Nodes which are connected and have the same degree as the normal degree are marked as vn+q2+1(ii) a q2 initial value is 0; x is 1 or 2; at each iteration, the values of q1 and q2 are equal.
6.7.4) judging q2>If Q2 is true, if yes, go to step 6.7.5), otherwise, make Q2 become Q2+1, and return to step 6.7.3); wherein, the node vn+Q2To be located at node vnAnd node vjxAnd node vjxNodes which are connected and have the same degree as the normal degree;
6.7.5) to be controlled by node vtNode vt+1…, node vt+Q1Node vnNode vn+1…, node vn+Q2Node vjxNode vmxForming a diagonal quadrilateral area by the triangular units which are the vertexes; namely, the diagonal quadrilateral area comprises a plurality of triangles which take the nodes as vertexes.
6.8) dividing the diagonal quadrilateral area into quadrilateral units. Each quadrilateral unit is composed of two adjacent triangular units. Wherein, the diagonal line of the quadrangle is the common edge of two adjacent triangular units.
6.9) deleting the current diagonal line of each quadrilateral unit, connecting the other diagonal line of each quadrilateral unit, and finishing the optimization.
7) And judging whether i is greater than or equal to n. If yes, updating the triangular mesh finite element model and returning to the step 5), otherwise, making i equal to i +1 and returning to the step 6).
8) And judging whether the current triangular mesh finite element model has a non-acute angle triangle, if so, turning to the step 9), and otherwise, outputting the current triangular mesh finite element model.
9) The angles of the triangular mesh cells are optimized and the process returns to step 8).
The steps for optimizing the angle of the triangular grid unit are as follows:
9.1) establishing a plurality of triangular mesh unit clusters based on the current triangular mesh finite element model. Any one triangular mesh unit cluster is a convex polygon formed by three adjacent triangular mesh units. At least one of the three triangular grid units is a non-acute-angle triangle.
And 9.2) respectively using the middle point of each edge of the triangular grid unit cluster as the center of a circle and using the length of the edge where the center of the circle is located as the diameter to make a circle.
9.3) inserting nodes G sampled randomly in effective positions in the non-intersected areas of all circles by utilizing a gap processing method. And connecting the node G with the vertex of the triangular mesh unit set to complete the angle optimization of the triangular mesh unit.
9.4) repeating the step 9.2) and the step 9.3) until all the triangular mesh unit clusters are processed.
Example 2:
a method for optimizing a two-dimensional triangular mesh finite element model comprises the following steps:
1) and acquiring an initial triangular mesh finite element model.
2) Establishing a node optimization objective function:
setting a node viDegree of (f)iNormal degree of FiWhen v isiWhen it is an internal node, F i6; when node viAs a boundary node, according to viAngle between adjacent 2 boundary lines
To calculate fiSo that v isiThe internal angle degree of each adjacent triangular unit is as close to pi/3 as possible. When F is changed, the critical angle
It should satisfy:
when F is presentiWhen 2,3, …, and 7, respectively, the following are obtained:
when in use
In different intervals, corresponding FiThe values of (a) are shown in fig. 2.
The degree of the irregularity of the boundary nodes needs to be measured by introducing proper evaluation indexes and defined
Degree deviation:
the irregularity degree Δ φ of the whole finite element grid is defined as the cumulative sum of the degree deviations of the individual nodes:
the formula (1) optimizes an objective function for the node degrees, and when the objective function is 0, the standard degree of the grid model is the highest. The process of optimizing the grid is to minimize the value of the objective function.
3) Obtaining the degrees of all nodes in the initial triangular mesh finite element model, and according to the difference of the positions of the nodes, specifying as follows:
I) defining the nodes positioned in the model as internal nodes, wherein the regular degree of the nodes is 6;
II) defining the nodes on the model boundary as boundary nodes, wherein the values of the normal degrees are shown in figure 2 according to different critical angles.
4) If the actual degree of the internal node is 3, the requirement that the regular degree of the internal node is 6 is not satisfied, as in the node B in fig. 3 (a). The node B is removed and the triangle units 1, 2,3 drawn from the node B are merged into one triangle unit, resulting in the result shown in fig. 3 (B).
If the actual degree of the internal node is 4, as shown in fig. 3(c), the requirement that the regular degree of the internal node is 6 is not satisfied. Then the node D is removed, and the triangle units 1, 2,3 and 4 led out by the node D are fused into a quadrangle I1I2I3I4. At this time, for quadrangle I1I2I3I4Subdivided, possibly connected to I1I3Or is connected to I2I4Since there cannot be non-acute triangles, the connection I2I4。
And aiming at other nodes which do not meet the normal degree, optimizing by adopting an edge commutation method. As shown in FIG. 4(a), if node TaIs FaThe actual degree of which is Fa-1; node TbIs FbThe actual degree of which is Fb+ 1; node TcIs of normal degree TcThe actual degree of which is Tc-1; node TdIs of normal degree TdThe actual degree of which is Td+ 1; the four nodes form a structure similar to a quadrangle, which is defined as a 'diagonal quadrangle structure', and the actual degree is Fa-1 node and actual degree TcThe nodes of-1 are located on the same diagonal of the quadrilateral, and the actual degree is FbNode of +1 and actual degree TdThe +1 nodes are located on the same diagonal of the quadrilateral, and the middle node Ta+1，Tb+1，…，Ta+n，Tb+nAll degrees of (a) are regular degrees of 6. At this time, the node T may be connectedb、TdThe connection lines in the direction (e.g. blue connection lines in FIG. 4 (a)) are reversed to reconnect the node Ta、TcThe connecting lines in the direction (e.g., red connecting lines in fig. 4 (b)).
The edge commutation of the triangular mesh shown in fig. 4 specifically comprises the following steps:
I) choose to be located at TaAnd TdBetween, in normal degrees and directly with TaThe connected nodes are marked as Ta+1；
II) selection at Ta+1And TdBetween, in normal degrees and directly with Ta+1The connected nodes are marked as Ta+2；
III) analogizing in turn until the node T is selecteda+nAnd T isa+nAnd TbDirectly connecting;
IV) similarly, selecting the position at TbAnd TcBetween, in normal degrees and directly with Ta+1,TbThe connected nodes are marked as Tb+1；
V) is selected to be at Tb+1And TcBetween, in normal degrees and directly with Ta+2,Tb+1The connected nodes are marked as Tb+2；
VI) repeating the steps until the node T is selectedb+nAnd T isb+nAnd TcDirectly connecting;
VII) commutation is described by connecting Ta+1Tb，Ta+2Tb+1……TdTb+nIs removed and then T is connectedaTb+1，Ta+ 1Tb+2……Ta+nTc(ii) a And n has a plurality of effective values, and in order to ensure the efficiency, the side is reversed in a mode of minimizing n.
5) Since the triangular mesh finite element model does not allow the existence of non-acute triangles, the optimization of the number of degrees of the nodes is completed, and then the optimization of the unit angles is needed. Defining a cluster, wherein the cluster needs to simultaneously satisfy the following three conditions:
I) the three triangles are adjacent in the grid;
II) a polygon formed by the three triangles is a convex polygon;
III) at least one of the three triangles is an obtuse angle or a right angle triangle.
The essence of the optimization of the unit angle isA new node is inserted in the cluster. As shown in FIG. 5(a), a triangle T3T7T8Triangle T2T3T7Triangle T2T6T7Forming a cluster by respectively making circles on the convex polygon T by taking the sides forming the convex polygon as diameters and the middle point of the boundary as the center of the circle2T3T6T7T8There is a non-intersecting region of all circles inside, the region forms a polygon S, a new point G sampled randomly is inserted as a node at an effective position inside the polygon S by adopting a gap processing technology, and the node is connected with a convex polygon T2T3T6T7T8The optimized mesh of the cluster is obtained, as shown in fig. 5 (b).
And traversing all the triangular units, and finishing the whole optimization if the non-acute-angle triangular unit does not exist.
Claims (6)
1. A method for optimizing a finite element model of a two-dimensional triangular mesh is characterized by comprising the following steps:
1) and acquiring the initial triangular mesh finite element model.
2) And establishing a node optimization model.
3) Determining a normal degree of each node, and writing a normal degree set F ═ F1，F2，…，FnIn (1) }; n is the number of nodes of the finite element model of the current triangular mesh;
4) calculating degrees of all nodes of the current triangular mesh finite element model, and writing a degree set f ═ f into the degree set1，f2，…，fnIn (1) };
5) inputting the degree set F and the regular degree set F of the initial triangular mesh finite element model into the node optimization model, and calculating the objective function of the node optimization model
If the objective function
If the reduction is not carried out, the step 8) is carried out, otherwise, the step 6) is carried out;
6) for node v of current triangular mesh finite element modeliOptimizing; i initial value is 1;
7) judging whether i is greater than or equal to n; if yes, updating the triangular mesh finite element model and returning to the step 5), otherwise, making i equal to i +1 and returning to the step 6);
8) judging whether a non-acute-angle triangle exists in the current triangular mesh finite element model, if so, turning to the step 9), and otherwise, outputting the current triangular mesh finite element model;
9) the angles of the triangular mesh cells are optimized and the process returns to step 8).
2. A method for optimizing a two-dimensional triangular mesh finite element model according to claim 1 or 2, wherein the step of determining the degree of regularity of each node is as follows:
1) judging node viType of (d);
when node viWhen the node is an internal node, the normal degree F is recordedi6, and go to step 3); when node viIf the node is a boundary node, turning to the step 2);
the internal nodes are nodes positioned in the initial triangular mesh finite element model; the boundary nodes are nodes positioned at the boundary of the finite element model of the initial triangular mesh; i initial value is 1;
2) obtaining a node viAngle between adjacent 2 boundary lines
When in use
Time, normal degree Fi＝2；
When in use
Time, normal degree Fi＝3；
When in use
Time, normal degree Fi＝4；
When in use
Time, normal degree Fi＝5；
When in use
Time, normal degree Fi＝6；
When in use
Time, normal degree Fi＝7；
At a certain node viAfter the normal degree, the step 3) is carried out;
3) and judging whether i is greater than or equal to n, if so, ending, otherwise, making i equal to i +1, and returning to the step 1).
3. The method of claim 1, wherein an objective function Δ φ of the node optimization model is as follows:
in the formula (f)iIs a node viThe degree of (d); fiIs a node viThe normal degree of (d); and delta phi is the irregularity degree of the current triangular mesh finite element model.
4. The method of claim 1, wherein the step of optimizing the nodes of the current triangular mesh finite element model comprises:
1) judging internal node viDegree of (f)iIf yes, ending the optimization; otherwise, turning to the step 2);
2) judging internal node viDegree of (f)iIf the result is 3, if so, the step is shifted to a step 3), and if not, the step is shifted to a step 4);
3) deleting internal nodes viAnd will be at internal node viThree triangular units as vertexes are fused into one triangular unit, and the optimization is finished;
4) judging internal node viDegree of (f)iIf yes, the step 5) is carried out, and if not, the step 6) is carried out;
5) deleting internal nodes viAnd will be at internal node viFour triangular units as vertexes are fused into a quadrilateral unit; connecting a diagonal line of the quadrilateral unit, dividing the quadrilateral unit into 2 acute-angle triangular units, and finishing optimization;
6) determining a node vtDegree of (f)tAnd normal degree Ft；
If number f of degreest＝Ft-1, respectively obtaining distance nodes vtNearest and satisfying degree fj1＝Fj1Node v of-1j1Distance internal node viNearest and satisfying degree fn1＝Fn1Node v of +1n1And distance node vj1Nearest and satisfying degree fm1＝Fm1Node v of +1m1(ii) a Connecting internal nodes vtNode vj1Node vn1Node vm1Forming a diagonal quadrilateral structure; wherein, the node vtAnd node vj1Are positioned on the same diagonal; node vn1And node vm1Are positioned on the same diagonal; node vtNode vn1Length of connecting line is less than or equal to node vtNode vm1The length of the connecting wire;
if number f of degreest＝Ft+1, respectively acquiring distance nodes vtNearest and satisfying degree fj2＝Fj2Node of +1vj2Distance node vtNearest and satisfying degree fn2＝Fn2Node v of-1n2And distance node vj2Nearest and satisfying degree fm2＝Fm2Node v of-1m2(ii) a Connecting node vtNode vj2Node vn2Node vm2Forming a diagonal quadrilateral structure; wherein, the node vtAnd node vj2Are positioned on the same diagonal; node vn2And node vm2Are positioned on the same diagonal; node vtNode vn2Length of connecting line is less than or equal to node vtNode vm2The length of the connecting wire;
7) determining a diagonal quadrilateral area, mainly comprising the following steps:
7.1) choose to locate at node vtAnd node vmxAnd node vt+q1Nodes which are connected and have the same degree as the normal degree are marked as vt+q1+1(ii) a q1 initial value is 0; x is 1 or 2;
7.2) judgment of q1>If Q1 is true, if yes, go to step 7.3), otherwise, make Q1 become Q1+1, and return to step 7.1); wherein, the node vt+Q1To be located at node vtAnd node vmxAnd node vmxNodes which are connected and have the same degree as the normal degree;
7.3) choose to locate at node vnAnd node vjxAnd node vn+q2And node vt+q1Nodes which are connected and have the same degree as the normal degree are marked as vn+q2+1(ii) a q2 initial value is 0; x is 1 or 2; q2 and q1 have equal values;
7.4) judgment of q2>If Q2 is true, if yes, go to step 7.5), otherwise, make Q2 become Q2+1, and return to step 7.3); wherein, the node vn+Q2To be located at node vnAnd node vjxAnd node vjxNodes which are connected and have the same degree as the normal degree;
7.5) to be controlled by node vt+q1Node vn+q2Node vjxNode vmxForming a diagonal quadrilateral area by the triangular units which are the vertexes; where q1 is ═ 0, 1,2，…，Q1]；q2＝[0,1,2，…，Q2]；
8) dividing the diagonal quadrilateral area into a plurality of quadrilateral units; each quadrilateral unit consists of two adjacent triangular units; wherein, the diagonal line of the quadrangle is the common edge of two adjacent triangular units;
9) deleting the current diagonal line of each quadrilateral unit, connecting the other diagonal line of each quadrilateral unit, and finishing the optimization.
5. The method of claim 1, wherein the step of optimizing the triangular mesh element angles comprises:
1) establishing a plurality of triangular mesh unit clusters based on the current triangular mesh finite element model; any one triangular mesh unit cluster is a convex polygon formed by three adjacent triangular mesh units; at least one of the three triangular grid units is a non-acute-angle triangle;
2) respectively taking the midpoint of each edge of the triangular grid unit cluster as the circle center, and taking the length of the edge where the circle center is located as the diameter to make a circle;
3) inserting nodes G sampled randomly into effective positions in the non-intersected areas of all circles by using a gap processing method; connecting the node G with the top point of the triangular mesh unit set to complete the angle optimization of the triangular mesh unit;
4) and repeating the step 2) and the step 3) until all the triangular grid unit clusters are processed.
6. A method for optimizing a finite element model of a two-dimensional triangular mesh according to claim 1, wherein the degree of the node represents the number of adjacent triangular elements with the node as one of the vertices.
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