Dimension reduction analysis method for finite element

Abstract

The invention belongs to the technical field of finite element analysis methods, and particularly relates to a dimension reduction analysis method for finite elements. Comprising the following steps: extracting a finite element model basic variable sequence to establish a basic model; collecting or designing samples to be analyzed to generate a finite element model; establishing a finite element mathematical model based on a structural finite element equilibrium equation; designing and determining a finite element variable fuzzy interval; and based on fuzzy variable set segmentation processing, fuzzy variable differential processing, fuzzy variable stiffness optimization based on Taylor series, finite element balance equation global stiffness optimization, displacement response determination and respective solution in a subset interval are involved. According to the method, the fuzzy variables involved in the finite element analysis process are expanded and simplified, derivative optimization calculation is performed through the actual physical relationship among the basic structure set parameters of the finite element equation, simplification and optimization in the calculation and analysis process can be better performed when the fuzzy parameters are involved, the operation efficiency is improved, and the calculation cost is reduced. And the time consumption of finite element analysis is reduced.

Description

Technical Field

[0001] The invention belongs to the technical field of finite element analysis methods, and in particular relates to a dimension reduction analysis method for finite elements. Background Art

[0002] Finite Element Analysis (FEA) is the process of predicting the behavior of an object based on finite element method (FEM) calculations. FEM is a mathematical method, and FEA is the interpretation of FEM results. FEA gives engineers a deep understanding of complex systems and structures, helping them make more informed design decisions. FEM breaks down complex systems into smaller, simpler parts (i.e., "units") based on mathematics. Next, it applies differential equations to each unit individually, using the power of the computer to divide it, and then solves the engineering problem. In the finite element analysis process. The analysis of the model structure response and the conditions under the contract are important contents of the finite element analysis process, but in actual engineering applications, the above analysis content often has modular parameters. These fuzzy parameters involve uncertain parameters, changing states, etc. in the analysis process. The processing of these fuzzy properties is a very difficult content that consumes computing resources and makes the calculation and analysis process complicated in the finite element analysis process. Summary of the invention

[0003] The purpose of the present invention is to provide a dimensionality reduction analysis method for finite elements based on actual needs to solve the problem of difficulty in finite element calculation and analysis in the presence of fuzzy uncertainty, to reduce the difficulty of finite element analysis calculation, to simplify the analysis process, and to improve analysis efficiency.

[0004] To achieve the above purpose, the present invention adopts the following technical solution.

[0005] A dimensionality reduction analysis method for finite element analysis comprises the following steps:

[0006] Step 1: Extract the basic variable sequence of the finite element model, design or collect the simulation variable sequence based on the simulation goal and method, and use the modeling program or the finite element program model generation end to establish the necessary finite element basic model;

[0007] Step 2: Collect or design samples to be analyzed, assign corresponding data to basic variable sequences, assign corresponding variables to simulation variable sequences according to the finite element simulation analysis task plan, and generate a finite element model;

[0008] Step 3: Establish a finite element mathematical model based on the basic principle of the structural finite element equilibrium equation A1: Ku = f; It refers to the global stiffness matrix used to describe the structure of the finite element model; Solution vectors for the finite element model nodes; is the load vector of each structural node during the finite element model simulation process, N d It refers to the model structure degree of freedom variable;

[0009] Step 4: Determine the finite element variable fuzzy interval according to the finite element task variable design. The fuzzy interval is used to characterize the variable value or variable range that is determined. Any variable or interval value in the fuzzy interval may be the optimal value or optimal interval in the finite element simulation design.

[0010] For the fuzzy variable set a=[a1,a2...a i ...a I ] T , the superscript T indicates the matrix transpose, where a i represents the i-th fuzzy variable. Based on the structural finite element equilibrium equation, the global stiffness matrix K(a′)∈K′ and the fuzzy load vector f(a)∈f′ of the fuzzy structure are defined. The corresponding finite element model node displacement vector u′ is the fuzzy solution vector. The finite element equilibrium equation involving fuzzy variables can be expressed as A2: K(a)u=f(a); the solution set satisfying the equilibrium equation can be expressed as A3: Ω={u|Ku=f}, K∈K′, f∈f′;

[0011] Step 5: Based on the segmentation process of the fuzzy variable set a′, the i-th fuzzy variable a in the fuzzy variable set is i The partitioning is performed to obtain J fuzzy variable subsets, namely a i =a i1 ∪a i2 ∪...a ij ...∪a iJ ; then in the solution space, the fuzzy variable subset a ij The interval can be expressed as A4: i∈[1,I],j∈[1,J];

[0012] The superscript R represents the interval, the superscript min represents the lower limit of the variable or variable set, and the superscript r represents the interval radius of the fuzzy variable or variable set in the solution space;

[0013] Get the fuzzy variable subset a in the solution space ij Interval midpoint It can be expressed as:

[0014]

[0015] Then the fuzzy variable subset a in the solution space ij Interval The degree of fuzziness can be expressed and approximated as:

[0016]

[0017] It can be seen that the fuzziness of the fuzzy variable subset is directly related to the number of subset divisions. The more subsets there are, the lower the fuzziness of the subsets.

[0018] For the fuzzy variable set a, it is expressed in the form of fuzzy variable subset combination as follows:

[0019]

[0020] k i ∈[1,2...J], represents the mth fuzzy variable of the i-th i subset; Φ q Represents the combination form of fuzzy variable subsets;

[0021] Step 6: The finite element equilibrium equation K(a)u=f(a) involving fuzzy variables can be expressed in differential form as A5:

[0022] Reorganizing the differential formula A5 to facilitate the calculation of partial derivatives yields A6: u(a) is the structural displacement response;

[0023] For the kth component u of the response vector of the finite element equilibrium equation K(a)u=f(a) involving fuzzy variables k ,u k is a function of u(a) A7: in is related to the fuzzy variable a i An irrelevant row vector has its kth element as 1 and the other elements as 0;

[0024] Then u k Partial derivative of the i-th fuzzy variable is a displacement vector, define the displacement vector and Then A8:

[0025]

[0026] Based on the symmetry of the stiffness matrix, we have Then the displacement response u(a) to the fuzzy variable a i The sensitivity can be expressed as:

[0027] in is the fuzzy variable a when the midpoint value changes i The corresponding displacement variables;

[0028] Step 7: Based on Taylor series, the global stiffness matrix K(a′) of the finite element equilibrium equation involving fuzzy variables is calculated in its fuzzy variable array a=[a1,a2...a i ...a I ] T The midpoint a mid Expand it to get A9: in K(a) at the midpoint a mid About fuzzy variable a i The j-order partial derivative; let the fuzzy variable array a=[a1,a2...a i ...a I ] T The vth fuzzy variable in remains unchanged, and the other fuzzy variables take their midpoint values ​​a i =a i mid ,(i≠v), then A9 is simplified to A10: Adding the above I equations together, we can get:

[0029]

[0030] Step 8: The global stiffness matrix K(a′) of the finite element equilibrium equation involving fuzzy variables can be rewritten as:

[0031] A11: K(a)=K mid +δK;

[0032] in It refers to the perturbation interval of the global stiffness matrix of the finite element equilibrium equation of fuzzy variables;

[0033] Then the fuzzy load vector of the finite element equilibrium equation involving fuzzy variables can be expressed as:

[0034] A12: f(a) = f mid +δf;

[0035] in Combining A11 and A12, the displacement response interval of the finite element equilibrium equation involving fuzzy variables is obtained as follows:

[0036]

[0037] in K mid The inverse matrix of ; using the vector sequence iteration method to represent A13, we get:

[0038]

[0039] Among them, the y-order displacement response of the finite element equilibrium equation involving fuzzy variables is:

[0040]

[0041] Determine the displacement component u of the displacement response area of ​​the finite element equilibrium equation involving fuzzy variables k The upper and lower limits of the y-order stage displacement response.

[0042] Step 9: Solve separately in the interval of fuzzy variable subsets, and obtain the fuzzy boundary of the corresponding solution space through interval union operation.

[0043] To further improve or implement the aforementioned dimensionality reduction analysis method for finite elements, the Abaqus mesh module is used to support global mesh control, global seed size setting, local seed size setting, and tetrahedral meshing and hexahedral meshing schemes.

[0044] Further improvement or specific implementation of the aforementioned dimensionality reduction analysis method for finite elements, the step 2 further includes: simplifying and cleaning non-critical features in the finite element model according to the final requirements of the simulation analysis task, completing the finite element model material setting and component type setting according to the finite element system configuration mode, and defining material properties, simulation loads and constraint definitions at the same time;

[0045] Further improvement or specific implementation of the aforementioned dimensionality reduction analysis method for finite elements, the simplification and cleaning include but are not limited to: using specific finite element model blocks to replace some non-critical model blocks or areas to simplify the model complexity of non-critical areas; replacing non-essential active connections with fixed connections to simplify the structural complexity during the simulation of non-essential areas; configuring global grid control variables, setting global seed size, local seed size, and selecting appropriate grid division form to divide the grid instead of analyzing grid quality.

[0046] Its beneficial effects are:

[0047] The present application expands and simplifies the fuzzy variables involved in the finite element analysis process, and derives optimization calculations through the actual physical relationship between the basic structural set parameters of the finite element equation. It can better simplify and optimize the calculation and analysis process when fuzzy parameters are involved, improve computational efficiency, and reduce the time consumption of finite element analysis. BRIEF DESCRIPTION OF THE DRAWINGS

[0048] Figure 1 It is a schematic diagram of the finite element analysis results for a certain displacement parameter based on the traditional finite element algorithm and the method based on the present application in the embodiment. DETAILED DESCRIPTION

[0049] The present invention is described in detail below in conjunction with specific embodiments.

[0050] The present application relates to a dimensionality reduction processing method for finite element analysis. The method is based on the finite element analysis processing method, and by deconstructing the actual physical relationship between finite element parameters and integrating the optimization operation process, the difficulty of parameter processing is reduced, the operation process and operation ideas are simplified, and the efficiency of the finite element algorithm is improved.

[0051] The basic process of the dimensionality reduction processing method for finite element analysis of the present application includes the following steps:

[0052] Step 1: Extract the basic variable sequence of the finite element model, design or collect the simulation variable sequence based on the simulation goal and method, and use the modeling program or the finite element program model generation end to establish the necessary finite element basic model;

[0053] Step 2: Collect or design samples to be analyzed, assign corresponding data to basic variable sequences, assign corresponding variables to simulation variable sequences according to the finite element simulation analysis task plan, and generate a finite element model;

[0054] According to the final requirements of the simulation analysis task, the non-critical features in the finite element model are simplified and cleaned up, and the finite element model material settings and component type settings are completed according to the finite element system configuration method. At the same time, material properties, simulation loads and constraint definitions are defined;

[0055] Including but not limited to: using specific finite element model blocks to replace some non-critical model blocks or areas to simplify the model complexity of non-critical areas; replacing non-essential active connections with fixed connections to simplify the structural complexity during the simulation of non-essential areas;

[0056] Configure global mesh control variables, set global seed size, local seed size, and select appropriate meshing methods to divide the mesh instead of analyzing mesh quality;

[0057] As a preferred method, the Abaqus mesh module can be used to support global mesh control, global seed size setting, local seed size setting, and tetrahedral meshing and hexahedral meshing schemes;

[0058] Step 3: Establish a finite element mathematical model based on the basic principle of the structural finite element equilibrium equation A1: Ku = f; It refers to the global stiffness matrix used to describe the structure of the finite element model; Solution vectors for the finite element model nodes; is the load vector of each structural node during the finite element model simulation process, N d It refers to the model structure degree of freedom variable;

[0059] Step 4: Determine the finite element variable fuzzy interval according to the finite element task variable design. The fuzzy interval is used to characterize the variable value or variable range that is determined. Any variable or interval value in the fuzzy interval may be the optimal value or optimal interval in the finite element simulation design.

[0060] For the fuzzy variable set a=[a1,a2...a i ...a I ] T , the superscript T indicates the matrix transpose, where a i represents the i-th fuzzy variable. Based on the structural finite element equilibrium equation, the global stiffness matrix K(a′)∈K′ and the fuzzy load vector f(a)∈f′ of the fuzzy structure are defined. The corresponding finite element model node displacement vector u′ is the fuzzy solution vector. The finite element equilibrium equation involving fuzzy variables can be expressed as A2: K(a)u=f(a); the solution set satisfying the equilibrium equation can be expressed as A3: Ω={u|Ku=f}, K∈K′, f∈f′;

[0061] Step 5: Based on the segmentation process of the fuzzy variable set a′, the i-th fuzzy variable a in the fuzzy variable set is i The partitioning is performed to obtain J fuzzy variable subsets, namely a i =a i1 ∪a i2 ∪...a ij ...∪a iJ ; then in the solution space, the fuzzy variable subset a ij The interval can be expressed as A4: i∈[1,I],j∈[1,J];

[0062] The superscript R represents the interval, the superscript min represents the lower limit of the variable or variable set, and the superscript r represents the interval radius of the fuzzy variable or variable set in the solution space;

[0063] Get the fuzzy variable subset a in the solution space ij Interval midpoint It can be expressed as:

[0064]

[0065] Then the fuzzy variable subset a in the solution space ij Interval The fuzziness degree can be expressed and approximated as

[0066]

[0067] It can be seen that the fuzziness of the fuzzy variable subset is directly related to the number of subset divisions. The more subsets there are, the lower the fuzziness of the subsets.

[0068] For the fuzzy variable set a, it is expressed in the form of fuzzy variable subset combination as follows:

[0069]

[0070] k i ∈[1,2...J], represents the mth fuzzy variable of the i-th i subset; Φ q Represents the combination form of fuzzy variable subsets;

[0071] Step 6: The finite element equilibrium equation involving fuzzy variables K(a)u=f(a) can be expressed in differential form as

[0072] Reorganizing the differential formula A5 to facilitate the calculation of partial derivatives yields A6: u(a) is the structural displacement response;

[0073] For the kth component u of the response vector of the finite element equilibrium equation K(a)u=f(a) involving fuzzy variables k ,u k is a function of u(a) A7: in is related to the fuzzy variable a i An irrelevant row vector has its kth element as 1 and the other elements as 0;

[0074] Then u k Partial derivative of the i-th fuzzy variable Q k T K(a) -1 is a displacement vector, define the displacement vector and Then A8:

[0075]

[0076] Based on the symmetry of the stiffness matrix, we have Then the displacement response u(a) to the fuzzy variable a i The sensitivity can be expressed as:

[0077] in is the fuzzy variable a when the midpoint value changes i The corresponding displacement variables;

[0078] Step 7: Based on Taylor series, the global stiffness matrix K(a′) of the finite element equilibrium equation involving fuzzy variables is calculated in its fuzzy variable array a=[a1,a2...a i ...a I ] T The midpoint a mid Expand it to get A9: in K(a) at the midpoint a mid About fuzzy variable a i The j-order partial derivative; let the fuzzy variable array a=[a1,a2...a i ...a I ] T The vth fuzzy variable in remains unchanged, and the other fuzzy variables take their midpoint values ​​a i =a i mid ,(i≠v), then A9 is simplified to A10:

[0079]

[0080] Adding the above I equations together, we can get:

[0081] Step 8: The global stiffness matrix K(a′) of the finite element equilibrium equation involving fuzzy variables can be rewritten as:

[0082] A11: K(a)=K mid +δK;

[0083] in It refers to the perturbation interval of the global stiffness matrix of the finite element equilibrium equation of fuzzy variables;

[0084] Then the fuzzy load vector of the finite element equilibrium equation involving fuzzy variables can be expressed as:

[0085] A12: f(a) = f mid +δf;

[0086] in Combining A11 and A12, the displacement response interval of the finite element equilibrium equation involving fuzzy variables is obtained as follows:

[0087]

[0088] in K mid The inverse matrix of ; using the vector sequence iteration method to represent A13, we get:

[0089]

[0090] Among them, the y-order displacement response of the finite element equilibrium equation involving fuzzy variables is:

[0091]

[0092] Determine the displacement component u of the displacement response area of ​​the finite element equilibrium equation involving fuzzy variables k The upper and lower limits of the y-order stage displacement response.

[0093] Step 9: Solve separately in the interval of fuzzy variable subsets, and obtain the fuzzy boundary of the corresponding solution space through interval union operation.

[0094] In order to illustrate the specific application effect of the present invention, a finite element analysis is performed on the hydrodynamic configuration of a twisted rudder of a certain type of power ship based on the existing traditional finite element analysis algorithm and the solution in this application. The basic parameter structure of the twisted rudder body is measured based on the actual size of the twisted rudder, such as Figure 1 The figure shows the schematic diagram of the finite element simulation results of the displacement of a certain area of ​​the twisted rudder under the two schemes. It can be seen from the figure that in the case of fuzzy parameters, the basic simulation results of the two are not much different, but relatively speaking, the calculation cost required for the calculation scheme based on the scheme of the present application is lower and the calculation speed is faster.

[0095] Finally, it should be noted that the above embodiments are only used to illustrate the technical solution of the present invention, rather than to limit the scope of protection of the present invention. Although the present invention has been described in detail with reference to the preferred embodiments, those skilled in the art should understand that the technical solution of the present invention can be modified or replaced by equivalents without departing from the essence and scope of the technical solution of the present invention.

Claims

1. A dimensionality reduction analysis method for finite element analysis, characterized in that: The steps include: Step 1: Extract the basic variable sequence of the finite element model, design or collect the simulation variable sequence based on the simulation goal and method, and use the modeling program or the finite element program model generation end to establish the necessary finite element basic model; Step 2: Collect or design samples to be analyzed, assign corresponding data to basic variable sequences, assign corresponding variables to simulation variable sequences according to the finite element simulation analysis task plan, and generate a finite element model; Step 3: Establish a finite element mathematical model based on the basic principle of the structural finite element equilibrium equation A1: Ku = f; It refers to the global stiffness matrix used to describe the structure of the finite element model; Solution vectors for the finite element model nodes; is the load vector of each structural node during the finite element model simulation process, N d It refers to the model structure degree of freedom variable; Step 4: Determine the finite element variable fuzzy interval according to the finite element task variable design. The fuzzy interval is used to characterize the variable value or variable range that is determined. Any variable or interval value in the fuzzy interval may be the optimal value or optimal interval in the finite element simulation design. For the fuzzy variable set a=[a1,a2...a i ...a I ] T , the superscript T indicates the matrix transpose, where a i represents the i-th fuzzy variable. Based on the structural finite element equilibrium equation, the global stiffness matrix K(a′)∈K′ and the fuzzy load vector f(a)∈f′ of the fuzzy structure are defined. The corresponding finite element model node displacement vector u′ is the fuzzy solution vector. The finite element equilibrium equation involving fuzzy variables can be expressed as A2: K(a)u=f(a); the solution set satisfying the equilibrium equation can be expressed as A3: Ω={u|Ku=f}, K∈K′, f∈f′; Step 5: Based on the segmentation process of the fuzzy variable set a′, the i-th fuzzy variable a in the fuzzy variable set is i The partitioning is performed to obtain J fuzzy variable subsets, namely a i =a i1 ∪a i2 ∪...a ij ...∪a iJ ; then in the solution space, the fuzzy variable subset a ij The interval can be expressed as A4: i∈[1,I],j∈[1,J]; The superscript R represents the interval, the superscript min represents the lower limit of the variable or variable set, and the superscript r represents the interval radius of the fuzzy variable or variable set in the solution space; Get the fuzzy variable subset a in the solution space ij Interval midpoint It can be expressed as: Then the fuzzy variable subset a in the solution space ij Interval The fuzziness degree can be expressed and approximated as It can be seen that the fuzziness of the fuzzy variable subset is directly related to the number of subset divisions. The more subsets there are, the lower the fuzziness of the subsets. For the fuzzy variable set a, it is expressed in the form of fuzzy variable subset combination as follows: k i ∈[1,2...J], represents the mth fuzzy variable of the i-th i subset; Φ q Represents the combination form of fuzzy variable subsets; Step 6: The finite element equilibrium equation K(a)u=f(a) involving fuzzy variables can be expressed in differential form as A5: Reorganizing the differential formula A5 to facilitate the calculation of partial derivatives yields A6: u(a) is the structural displacement response; For the kth component u of the response vector of the finite element equilibrium equation K(a)u=f(a) involving fuzzy variables k ,u k is a function of u(a) A7: in is related to the fuzzy variable a i An irrelevant row vector has its kth element as 1 and the other elements as 0; Then u k Partial derivative of the i-th fuzzy variable is a displacement vector, define the displacement vector and Then A8: Based on the symmetry of the stiffness matrix, we have Then the displacement response u(a) to the fuzzy variable a i The sensitivity can be expressed as: in is the fuzzy variable a when the midpoint value changes i The corresponding displacement variables; Step 7: Based on Taylor series, the global stiffness matrix K(a′) of the finite element equilibrium equation involving fuzzy variables is calculated in its fuzzy variable array a=[a1,a2...a i ...a I ] T The midpoint a mid Expand it to get A9: in K(a) at the midpoint a mid About fuzzy variable a i The j-order partial derivative of Let the fuzzy variable array a=[a1,a2...a i ...a I ] T The vth fuzzy variable in remains unchanged, and the other fuzzy variables take their midpoint values ​​a i =a i mid ,(i≠v), then A9 is simplified to A10: Adding the above I equations together, we can get: Step 8: The global stiffness matrix K(a′) of the finite element equilibrium equation involving fuzzy variables can be rewritten as: A11:K(a)=K mid +δK; in It refers to the perturbation interval of the global stiffness matrix of the finite element equilibrium equation of fuzzy variables; Then the fuzzy load vector of the finite element equilibrium equation involving fuzzy variables can be expressed as: A12:f(a)=f mid +δf; in Combining A11 and A12, the displacement response interval of the finite element equilibrium equation involving fuzzy variables is obtained as follows: A13： in K mid The inverse matrix of ; using the vector sequence iteration method to represent A13, we get: A14： Among them, the y-order displacement response of the finite element equilibrium equation involving fuzzy variables is: A15： Determine the displacement component u of the displacement response area of ​​the finite element equilibrium equation involving fuzzy variables k The upper and lower limits of the y-order stage displacement response. Step 9: Solve separately in the interval of fuzzy variable subsets, and obtain the fuzzy boundary of the corresponding solution space through interval union operation.

2. A dimensionality reduction analysis method for finite element analysis according to claim 1, characterized in that: The Abaqus mesh module is used to support global mesh control, global seed size setting, local seed size setting, and tetrahedral meshing and hexahedral meshing schemes.

3. A dimensionality reduction analysis method for finite element analysis according to claim 1, characterized in that: The step 2 also includes: simplifying and cleaning non-critical features in the finite element model according to the final requirements of the simulation analysis task, completing the finite element model material setting and component type setting according to the finite element system configuration method, and defining material properties, simulation loads and constraint definitions.

4. A dimensionality reduction analysis method for finite element analysis according to claim 3, characterized in that: The simplification and cleaning include but are not limited to: using specific finite element model blocks to replace some non-critical model blocks or areas to simplify the model complexity of non-critical areas; replacing non-essential active connections with fixed connections to simplify the structural complexity of non-essential areas during simulation; configuring global mesh control variables, setting global seed size, local seed size, and selecting appropriate mesh division form to divide the mesh instead of analyzing mesh quality.

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