Ten-node tetrahedron element strain reconstruction and hourglass control method
By dividing the ten-node tetrahedral element into four hexahedrons and constructing a novel hourglass pattern, the problems of volume self-locking and pressure checkerboard oscillation are solved, improving computational stability and accuracy, and making it suitable for simulation analysis of complex geometric structures.
Patent Information
- Application Number
- CN202511504637.9
- Authority / Receiving Office
- CN · China
- Patent Type
- Applications(China)
- Current Assignee / Owner
- Filing Date
- 2025-10-21
- Publication Date
- 2025-11-21
AI Technical Summary
Existing ten-node tetrahedral elements cannot effectively eliminate volumetric self-locking and pressure checkerboard oscillation problems in strain analysis, especially when dealing with approximately incompressible materials, resulting in poor computational stability.
By dividing the ten-node tetrahedral element into four hexahedral elements and constructing a new hourglass pattern, including hourglass force, hourglass tangent matrix and element hourglass stiffness matrix, a new stiffness matrix is constructed using the nature of hourglass deformation to control the self-locking phenomenon.
It significantly improves computational stability and accuracy, reduces volumetric and shear-locking phenomena, enhances computational efficiency, and is suitable for simulation analysis of complex geometries.
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Figure CN120995794A_ABST
Abstract
Description
TECHNICAL FIELD
[0001] The present application relates to the technical field of numerical simulation analysis in engineering, and particularly relates to a ten-node tetrahedral element strain reconstruction and hourglass control method. BACKGROUND
[0002] In the field of structural finite element analysis, tetrahedral elements have become an indispensable tool for discretization of engineering models due to their natural adaptability to complex geometric topologies. Early tetrahedral elements were mostly linear four-node elements, which could handle complex curved surfaces and hole structures, but had significant inherent defects: the stress within the element was constant and could not accurately describe the continuous stress field; shear locking phenomenon was easily induced in bending and contact problems; the overall stiffness matrix was "overly stiff", leading to insufficient displacement solution and poor calculation accuracy. These problems seriously restricted the application value of tetrahedral elements in key industrial scenarios. To break through the above limitations, ten-node quadratic tetrahedral elements emerged. The element introduces additional nodes at the midpoint of each edge of the tetrahedron, extending the displacement function from linear to quadratic polynomial. This change significantly improves the element's ability to describe stress gradients: it can simulate linear stress distribution and effectively alleviate shear locking; with the same number of degrees of freedom, its calculation accuracy is comparable to that of twenty-node hexahedral elements, while the system matrix has smaller bandwidth, making the solution speed increase by 15%-30%. For this reason, mainstream CAE platforms such as ANSYS Workbench and Abaqus have listed it as the preferred element for complex models, and it is widely used in industrial models with complex geometric characteristics such as engine blocks, aircraft engine blades, and biological implants.
[0003] The competitiveness of ten-node tetrahedral elements comes from three technical breakthroughs: first, geometric inclusivity innovation. Traditional hexahedral grids often need to be manually cut to match the sweep path when dividing highly irregular structures such as engine block water jackets and turbine blade cooling channels, which takes several hours to several days. However, tetrahedral grids can automatically divide any topology through algorithms such as Delaunay triangulation without the need for geometric simplification. For example, a certain type of high-pressure turbine disk for an aircraft engine contains 78 cooling holes and curved mortise and tenon joints. It takes 32 hours to divide the hexahedral grid, while the tetrahedral element only takes 9 minutes to complete the discretization, significantly speeding up the iteration process.
[0004] Second, the synergy optimization of precision and efficiency. Early research suggests that tetrahedral element precision is inherently lower than hexahedral. However, through comparative tests of typical working conditions such as bending, twisting, and contact, this perception has been overturned: the secondary tetrahedral and hexahedral elements are comparable in terms of precision and CPU time. The reason is that the system matrix of the ten-node tetrahedral element has a smaller bandwidth (fewer adjacent node connections), and the optimization of the sparse matrix iterative solver reduces memory requirements. A certain vehicle crash simulation case shows that, under the same degrees of freedom, the ten-node tetrahedral model takes 28% less time to solve and 35% less peak memory usage than the twenty-node hexahedral model.
[0005] Third, the engineering value of adaptive capability. In strong nonlinear problems such as crack propagation and plastic deformation, the ten-node tetrahedral element supports dynamic redivision based on stress gradients. For example, in pressure vessel fatigue analysis, the software can automatically insert high-order nodes at the crack tip based on the Von Mises stress error threshold, keeping the stress intensity factor calculation error within 5%. In contrast, the re-meshing of hexahedral grids requires the reconstruction of topological mapping relationships, resulting in exponential growth in computational cost.
[0006] While the standard ten-node tetrahedral element can solve the shear self-locking problem, it cannot eliminate volume self-locking and pressure checkerboard oscillation, and it also introduces a new problem of node reaction force oscillation. It is worth noting that the problems of volume self-locking and pressure checkerboard oscillation not only occur in rubber-like materials but also widely exist in materials with approximate incompressibility, such as viscoelastic materials and elastic-plastic materials. The key challenge is that simply refining the mesh cannot solve these problems. Based on this technical background, research on high-precision tetrahedral elements for accurate analysis of approximately incompressible materials has been an active research focus in computational mechanics to date. SUMMARY
[0007] The purpose of the present application is to provide a ten-node tetrahedral element strain reconstruction and hourglass control method to solve the problems of volume self-locking and pressure checkerboard oscillation that cannot be eliminated by current ten-node elements.
[0008] The present application provides a ten-node tetrahedral element strain reconstruction and hourglass control method, comprising: Step one, constructing four hexahedral elements based on ten-node tetrahedral elements; wherein the tetrahedral element is divided into four hexahedrons based on the center point of the element, the midpoint of the edge, and the center point of the face of the tetrahedral element; Step two, constructing the equivalent strain matrix of the ten-node tetrahedral element based on the four hexahedrons; Step three, constructing a new hourglass mode; wherein the hourglass mode is obtained by determining the difference between the center point of the tetrahedral element and the actual position, and the hourglass force, hourglass tangent matrix, element hourglass stiffness, and internal force and stiffness matrix in the element are obtained.
[0009] Further, in step one, in the ten-node tetrahedral element, node one to node four are the vertices of the ten-node tetrahedral element, node five to node ten are the midpoints of the edges of the ten-node tetrahedral element, node eleven is the center point of the tetrahedral element selected from the interior of the tetrahedral element, node twelve to node fifteen are the face center points selected from the four faces of the tetrahedral element respectively, and the coordinates of each node are as follows: (1) wherein, ... the coordinates of node one to node ten are respectively, the coordinate of node eleven is, the coefficients obtained according to arrangement, , .
[0010] Further, the coordinates of the face center points selected from the four faces of the tetrahedral element respectively are as follows: (2) wherein, ... the coordinates of node twelve to node fifteen are respectively, , .
[0011] Further, in step two, for the assembly of the tetrahedral element, the linear constraint on the middle node of the tetrahedral element is used to create the effective shape function of node one to node eleven; considering the current position of a point in the first hexahedron of the tetrahedral element: (3) wherein, is the interpolation function of the hexahedron, I = 1, 5, 7, 8, 11, 12, 13, 15; by combining formula (1) and (2), we have: (4) after arrangement, we have: (5) wherein, is the effective shape function; (6) The derivative of the uniform strain space is expressed as: (7) (8) wherein, is the volume corresponding to the hexahedron.
[0012] Furthermore, in step three, the difference between the center point of the tetrahedron determined by equation (1) and its actual position is... Result in hourglass mode: (9) hourglass vector The components 1-10 are determined by formula (1), and in addition, .
[0013] Furthermore, in step three, the hourglass force is expressed as: (10) in, It is the hourglass force. l It is the radius of the circumscribed sphere of the tetrahedron. , It is the radius of the inscribed ball. It is the linear hourglass stiffness, and its magnitude is Young's modulus or greater. and These are the control parameters for a nonlinear hourglass. ,and .
[0014] Furthermore, in step three, (11) in, The hourglass vector of the unit; The hourglass tangent matrix is represented as: (12) in, This is the hourglass tangent matrix.
[0015] Furthermore, in step three, the stiffness of the hourglass element is: (13) in, Let be the stiffness of the hourglass unit.
[0016] Furthermore, in step three, the internal force and stiffness matrices of the element are finally obtained: (14) (15) in, The internal forces in this unit, Here is the stiffness matrix of this element. For stress vectors, It is a unit cell.
[0017] This invention offers the following advantages: A method for strain reconstruction and hourglass control using ten-node tetrahedral elements is proposed. Based on ten-node tetrahedral elements, the ten-node tetrahedron is divided into four hexahedrons for strain reconstruction based on the element center point, edge midpoint, and face center point. Simultaneously, to save computational resources, a novel hourglass stiffness is constructed based on the nature of hourglass deformation, forming a novel calculation method for ten-node tetrahedral elements. Compared to traditional ten-node tetrahedral elements, it avoids volumetric and shear locking phenomena, resulting in higher accuracy. Compared to calculating four hexahedral elements, the novel hourglass control method proposed in this invention significantly improves computational efficiency while maintaining computational accuracy. Attached Figure Description
[0018] To more clearly illustrate the technical solution of the present invention, the drawings used in the embodiments will be briefly introduced below. Obviously, those skilled in the art can obtain other drawings based on these drawings without creative effort.
[0019] Figure 1 A schematic diagram of a ten-node tetrahedral element; Figure 2 This is a schematic diagram of four hexahedrons. Detailed Implementation
[0020] To make the objectives, technical solutions, and advantages of this invention clearer, the technical solutions of this invention will be clearly and completely described below in conjunction with specific embodiments and corresponding drawings. Obviously, the described embodiments are only a part of the embodiments of this invention, and not all of them. Based on the embodiments of this invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of this invention. The technical solutions provided by various embodiments of this invention will be described in detail below with reference to the accompanying drawings.
[0021] Please see Figure 1 This invention provides a method for strain reconstruction and hourglass control of ten-node tetrahedral elements, comprising: Step 1: Construct four hexahedral elements based on ten-node tetrahedral elements; wherein, the tetrahedral elements are divided into four hexahedral elements based on the element center point, edge midpoint, and face center point of the tetrahedral elements.
[0022] In a ten-node tetrahedral element, nodes one through four are the vertices of the ten-node tetrahedral element, nodes five through ten are the midpoints of the edges of the ten-node tetrahedral element, node eleven is the center point of the tetrahedral element selected from its interior, and nodes twelve through fifteen are the center points of the faces selected from the four faces of the tetrahedral element. Please refer to [link / reference]. Figure 1Each node is represented by a corresponding Arabic numeral; for example, node one is represented as "1". Based on a conventional ten-node tetrahedral element, point 11 is selected from inside the tetrahedron, and points 12, 13, 14, and 15 are selected from faces 1-7-3-6-2-5, 1-5-2-9-4-8, 2-6-3-10-4-9, and 1-8-4-10-3-7, respectively. The coordinates of each node are as follows: (1) in, ... These are the coordinates of nodes one through ten. For the coordinates of node eleven, In accordance with The coefficients obtained from the sorting process , .
[0023] The coordinates of the center points of the four faces of the tetrahedral element are as follows: (2) in, ... These are the coordinates of nodes twelve through fifteen. , .
[0024] Step 2: Construct the equivalent transformation matrix of a ten-node tetrahedron based on the four hexahedrons.
[0025] Figure 2 Figures (a) to (d) show four hexahedral diagrams. Specifically, for the assembly of tetrahedral elements, effective shape functions for nodes one through eleven are created using linear constraints on the nodes in the middle of the tetrahedron. Consider the current position of a point in the first hexahedron of the tetrahedron: (3) in, It is a hexahedral interpolation function, I=1,5,7,8,11,12,13,15; Combining formulas (1) and (2), we get: (4) Summarized as follows: (5) in, It is an effective shape function; (6) The spatial derivative of uniform strain is expressed as: (7) (8) in, It corresponds to the volume of a hexahedron.
[0026] Step 3: Construct a new hourglass pattern; wherein, the hourglass pattern is obtained based on the difference between the center point of the determined tetrahedral element and the actual position, and the hourglass force, hourglass tangent matrix, element hourglass stiffness, and internal force and stiffness matrix in the element are obtained.
[0027] On the one hand, for the newly added When a node is constrained, its association with other nodes on the tetrahedron increases the self-locking behavior of the tetrahedral elements; conversely, releasing constraints on a node increases the number of hourglass patterns. Taking all factors into consideration, when constraining nodes on a tetrahedron... The displacement of an intermediate node 11, due to the large number of associated nodes, significantly increases volume and shear locking. Retaining it as an additional degree of freedom can greatly alleviate the locking phenomenon. Therefore, in this invention, the intermediate node 11 is allowed to be unconstrained.
[0028] However, this approach leads to hourglass mode penalties, meaning that a given deformation gradient may involve different deformation modes. Therefore, it is necessary to increase constraints on different nodes to reduce hourglass modes. Hourglass control could be applied to the four hexahedrons divided by the 10-node tetrahedron to suppress hourglassing. However, to improve computational efficiency, this invention introduces a more computationally efficient hourglass control method, based on the difference between the tetrahedron center point and its actual position determined by equation (1). Result in hourglass mode: (9) hourglass vector The components 1-10 are determined by formula (1), and in addition, .
[0029] The hourglass force is represented as: (10) in, It is the hourglass force. l It is the radius of the circumscribed sphere of the tetrahedron. , It is the radius of the inscribed ball. It is the linear hourglass stiffness, and its magnitude is Young's modulus or greater. and These are the control parameters for a nonlinear hourglass. ,and .
[0030] (11)
[0031] in, The hourglass vector of the unit; The hourglass tangent matrix is represented as: (12) in, This is the hourglass tangent matrix.
[0032] The stiffness of the hourglass unit is: (13) in, Let be the stiffness of the hourglass unit.
[0033] Finally, the internal force and stiffness matrices of this element are obtained: (14) (15) in, The internal forces in this unit, Here is the stiffness matrix of this element. For stress vector, It is a unit cell.
[0034] As can be seen from the above embodiments, the present invention constructs four hexahedral elements through the face nodes and internal nodes of a ten-node tetrahedron, and constructs a novel shape function based on the hexahedral elements. It also proposes a novel hourglass control method, which greatly improves the computational efficiency and significantly overcomes the volume self-locking and hourglass phenomenon, thus greatly improving the computational stability. The novel hourglass control method of the present invention makes full use of the essential problem of the hourglass phenomenon and is a very effective method for handling hourglass control, which is applicable to implicit and explicit calculations in structural simulation.
[0035] The embodiments of the present invention described above do not constitute a limitation on the scope of protection of the present invention.
Claims
1. A method for strain reconstruction and hourglass control of ten-node tetrahedral elements, characterized in that, include: Step 1: Construct four hexahedral elements based on ten-node tetrahedral elements; wherein, the tetrahedral elements are divided into four hexahedral elements based on the element center point, edge midpoint, and face center point of the tetrahedral elements; Step 2: Construct the equivalent transformation matrix of a ten-node tetrahedron based on the four hexahedrons; Step 3: Construct a new hourglass pattern; wherein, the hourglass pattern is obtained based on the difference between the determined center point of the tetrahedral element and the actual position, and the hourglass force, hourglass tangent matrix, element hourglass stiffness, and internal force and stiffness matrix in the element are obtained. In step one, in the ten-node tetrahedral element, nodes one to four are the vertices of the ten-node tetrahedral element, nodes five to ten are the midpoints of the edges of the ten-node tetrahedral element, node eleven is the center point of the tetrahedral element selected from inside the tetrahedral element, and nodes twelve to fifteen are the center points of the faces selected from the four faces of the tetrahedral element respectively. The coordinates of each node are as follows: (1) in, ... These are the coordinates of nodes one through ten. For the coordinates of node eleven, In accordance with The coefficients obtained from the sorting process , ; The coordinates of the center points of the four faces of the tetrahedral element are as follows: (2) in, ... These are the coordinates of nodes twelve through fifteen. , ; In step two, for the assembly of the tetrahedral elements, effective shape functions for nodes one through eleven are created using linear constraints on the nodes in the middle of the tetrahedron; consider the current position of a point in the first hexahedron of the tetrahedron: (3) in, It is a hexahedral interpolation function, I=1,5,7,8,11,12,13,15; Combining formulas (1) and (2), we get: (4) Summarized as follows: (5) in, It is an effective shape function; (6) The spatial derivative of uniform strain is expressed as: (7) (8) in, It corresponds to the volume of a hexahedron.
2. The method for strain reconstruction and hourglass control of a ten-node tetrahedral element as described in claim 1, characterized in that, In step three, the difference between the center point of the tetrahedron determined by equation (1) and its actual position is... Result in hourglass mode: (9) hourglass vector The components 1-10 are determined by formula (1), and in addition, .
3. The method for strain reconstruction and hourglass control of a ten-node tetrahedral element as described in claim 2, characterized in that, In step three, the hourglass force is expressed as: (10) in, It is the hourglass force. l It is the radius of the circumscribed sphere of the tetrahedron. , It is the radius of the inscribed ball. It is the linear hourglass stiffness, and its magnitude is Young's modulus or greater. and These are the control parameters for a nonlinear hourglass. ,and .
4. The method for strain reconstruction and hourglass control of a ten-node tetrahedral element as described in claim 3, characterized in that, In step three, (11) in, The hourglass vector of the unit; The hourglass tangent matrix is represented as: (12) in, This is the hourglass tangent matrix.
5. The method for strain reconstruction and hourglass control of a ten-node tetrahedral element as described in claim 4, characterized in that, In step three, the stiffness of the hourglass element is: (13) in, Let be the stiffness of the hourglass unit.
6. The method for strain reconstruction and hourglass control of a ten-node tetrahedral element as described in claim 5, characterized in that, In step three, the internal force and stiffness matrices of the element are finally obtained: (14) (15) in, The internal forces in this unit, Here is the stiffness matrix of this element. For stress vectors, It is a unit cell.