A Dimensionality Reduction Analysis Method for Finite Element Methods

Abstract

The present invention belongs to the technical field of finite element analysis methods, and in particular relates to a dimensionality reduction analysis method for finite elements. It includes the following steps: extracting the basic variable sequence of the finite element model 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 the structural finite element equilibrium equation; designing and determining the fuzzy interval of the finite element variables; based on the fuzzy variable set segmentation processing, involving fuzzy variable differential processing, based on Taylor series fuzzy variable stiffness optimization, finite element equilibrium equation global stiffness optimization, determination of displacement response, and solving separately in the subset interval. This application expands and simplifies the fuzzy variables involved in the finite element analysis process, and derives the optimization calculation through the actual physical relationship between the basic structural set parameters of the finite element equation, which can better simplify and optimize the calculation and analysis process when fuzzy parameters are involved, improve the operation efficiency, and reduce the time consumption of finite element analysis.

Description

Technical Field

[0001] The present 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 a process that predicts 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 mathematically decomposes complex systems into smaller, simpler parts (i.e., "cells"). Next, it applies differential equations to each cell individually, using the power of computers to divide and solve engineering problems. During the finite element analysis process, the analysis of the model's structural response and the conditions under which it is performed are important aspects of the finite element analysis process. However, in actual engineering applications, the above analysis content often contains modular parameters. These fuzzy parameters involve uncertain parameters, changing states, etc. during the analysis process. The processing of these fuzzy properties is a difficult and computationally intensive part of the finite element analysis process, making the analysis process complex. 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 under the existence of fuzzy uncertainty, to reduce the difficulty of finite element analysis calculation, simplify the analysis process, and improve the analysis efficiency.

[0004] To achieve the above objectives, the present invention adopts the following technical solutions.

[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 the corresponding data to the basic variable sequence, assign the corresponding variables to the simulation variable sequence 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; Refers to the global stiffness matrix used to describe the structure of the finite element model; Solution vector for the finite element model node; 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 based on the finite element task variable design. The fuzzy interval is used to represent 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 represents the matrix transpose, where a i Denotes 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 that satisfies 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 divided into i The fuzzy variables are divided into J 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] Where superscript R represents the interval, superscript min represents the lower limit of the variable or variable set, and 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 fuzziness level 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 subset.

[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 subsets; Φ q Represents the combination form of fuzzy variable subsets;

[0021] Step 6: The finite element equilibrium equation involving fuzzy variables K(a)u=f(a) 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 rest 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) is related 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 of 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 express 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 the problem 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 mesh partitioning and hexahedral mesh partitioning schemes.

[0044] Further improvement or specific implementation of the aforementioned dimensionality reduction analysis method for finite elements, said 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 constraints;

[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 rather than analyze grid quality.

[0046] Its beneficial effects are:

[0047] This application expands and simplifies the fuzzy variables involved in the finite element analysis process, and performs derivative optimization calculations based on 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 with reference to specific embodiments.

[0050] The present application relates to a dimensionality reduction processing method for finite element analysis. This method is based on the finite element analysis processing method. 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 in this 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 the corresponding data to the basic variable sequence, assign the corresponding variables to the simulation variable sequence 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. The finite element model material settings and component type settings are completed according to the finite element system configuration method. At the same time, the 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 movable 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 mesh division method 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; Refers to the global stiffness matrix used to describe the structure of the finite element model; Solution vector for the finite element model node; 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 based on the finite element task variable design. The fuzzy interval is used to represent 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 represents the matrix transpose, where a i Denotes 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 that satisfies 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 divided into i The fuzzy variables are divided into J 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] Where superscript R represents the interval, superscript min represents the lower limit of the variable or variable set, and 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 subset.

[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 subsets; 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 rest 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) is related 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 of 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 express 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 the problem separately in the interval of fuzzy variable subsets, and obtain the fuzzy boundary of the corresponding solution space through interval union operation.

[0094] To illustrate the specific application effects 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 based on the actual size of the twisted rudder, such as Figure 1 The following is a schematic diagram of the finite element simulation results of the displacement of a certain area of ​​the twisted rudder under the two schemes. As can be seen from the figure, when fuzzy parameters are included, the basic simulation results of the two schemes are not much different. However, the calculation cost of the solution based on the present application is relatively lower and the calculation speed is faster.

[0095] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions 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 solutions of the present invention may be modified or replaced by equivalents without departing from the essence and scope of the technical solutions 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 finite element model variable sequence, 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 finite element basic model; Step 2: Collect or design samples to be analyzed, assign the corresponding data to the basic variable sequence, assign the corresponding variables to the simulation variable sequence 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 structural finite element equilibrium equation A1: The basic principle is to establish the finite element mathematical model; Refers to the global stiffness matrix used to describe the structure of the finite element model; Solution vector for the finite element model node; is the load vector of each structural node during the finite element model simulation process, It refers to the model structure degree of freedom variable; Step 4: Determine the finite element variable fuzzy interval based on the finite element task variable design. The fuzzy interval is used to represent 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 designed in the finite element analysis calculation process , superscript represents the matrix transpose, where Indicates the Fuzzy variables, based on the structural finite element equilibrium equation, define the global stiffness matrix of the fuzzy structure and the fuzzy load vector , the corresponding finite element model node displacement vector is the fuzzy solution vector, and the finite element equilibrium equation involving fuzzy variables can be expressed as A2: ; The solution set that satisfies the equilibrium equation can be expressed as A3: , , , ; Step 5: Based on fuzzy variable set The fuzzy variables are concentrated in the first Fuzzy variables Split to obtain A subset of fuzzy variables, namely ; then the fuzzy variable subset in the solution space The interval can be expressed as A4: ; ; The superscript Indicates interval, superscript Indicates the lower limit of a variable or set of variables, superscript Indicates the interval radius of the fuzzy variable or variable set in the solution space; Get the subset of fuzzy variables in the solution space interval midpoint It can be expressed as: ； Then the fuzzy variable subset in the solution space 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 subset. For the fuzzy variable set , which is expressed in the form of fuzzy variable subset combination as follows: ； , Indicates the The fuzzy variable subsets; Represents the combination form of fuzzy variable subsets; Step 6: Finite element equilibrium equations involving fuzzy variables It can be expressed as A5 in differential form: ; Reorganizing the differential formula A5 to facilitate the calculation of partial derivatives yields A6: ; is the structural displacement response; For finite element equilibrium equations involving fuzzy variables The response vector of Quantity , yes Function A7: ;in is related to fuzzy variables Independent row vectors, whose elements are 1, and the other elements are 0; but About Partial derivatives of fuzzy variables ; 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 For fuzzy variables The sensitivity can be expressed as: ; in Fuzzy variables when midpoint value changes the corresponding displacement variables; Step 7: Global stiffness matrix of finite element equilibrium equation involving fuzzy variables based on Taylor series In its fuzzy variable array midpoint Expand it to get A9: ;in for At the midpoint About fuzzy variables of order partial derivatives; Let the fuzzy variable array The first The fuzzy variables remain unchanged, and the other fuzzy variables take their midpoint values , then A9 is simplified to A10: ; The fuzzy variable array By adding the expanded simplified equations of the elements, we can get: ; Step 8: Global stiffness matrix of the finite element equilibrium equation involving fuzzy variables Can be rewritten as: A11： ； 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： ； in ; Combining A11 and A12, the displacement response interval of the finite element equilibrium equation involving fuzzy variables is: A13： ； in for The inverse matrix of ; using the vector sequence iteration method to express A13, we get: A14： ； Among them, the finite element equilibrium equation involving fuzzy variables is The displacement response of the first order stage is: A15： ； Determining the displacement components of the displacement response region of the finite element equilibrium equation involving fuzzy variables of Upper and lower limits of the step-stage displacement response; Step 9: Solve the problem 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: using finite element model blocks to replace some non-critical model blocks or areas to simplify the model complexity of non-critical areas; replacing active connections with fixed connections to simplify the structural complexity during regional simulation; configuring global grid control variables, setting global seed size, local seed size, and selecting an appropriate grid division form to divide the grid rather than analyze the grid quality.

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