JP7831125B2 - Analysis apparatus and analysis method - Google Patents

Description

This disclosure relates to an analysis device and an analysis method.

In product development, there is a need to predict the stress and deformation behavior that occurs in components. As a method for predicting the stress and deformation behavior that occurs in components, finite element analysis, which utilizes the finite element method, is used, for example, as disclosed in Patent Document 1.

Japanese Patent Publication No. 2019-160198

In finite element analysis, it is necessary to divide the model into multiple meshes (i.e., multiple elements). As the model's geometry becomes more complex, the number of meshes increases exponentially, leading to increased computational costs. Therefore, reducing the computational cost in finite element analysis is highly desirable.

The purpose of this disclosure is to provide an analysis apparatus and analysis method capable of reducing computational costs in finite element analysis.

To solve the above problems, the analysis apparatus of this disclosure comprises: a specification unit that identifies a first stiffness matrix, which is the stiffness matrix of the first model, based on the analysis results of a finite element analysis using a first model including a first base portion and a specific shape portion; a derivation unit that derives a third stiffness matrix, which is the stiffness matrix of the specific shape portion, by subtracting a second stiffness matrix, which is the stiffness matrix of the first base portion, from the first stiffness matrix; and a generation unit that generates a second model in which shell elements, set as the stiffness matrix of the third stiffness matrix, are attached to the second base portion.

The specific part may be determined based on the analysis results of a finite element analysis using a third model obtained by removing the specific shape part from the first model, thereby determining the second stiffness matrix.

The first base portion has the basic shape of the workpiece, and the specific shape portion may have a more complex shape than the first base portion.

The specific shaped portion may have the shape of a machining mark formed on the workpiece.

The stiffness matrix may include a matrix that defines the relationship between in-plane stress and in-plane strain.

The stiffness matrix may include a matrix that defines the relationship between moment and curvature.

The stiffness matrix may include a matrix that defines the relationship between out-of-plane shear stress and out-of-plane shear strain.

To solve the above problems, the analysis method of this disclosure includes the steps of: identifying a first stiffness matrix, which is the stiffness matrix of the first model, based on the analysis results of a finite element analysis using a first model including a first base portion and a specific shape portion; deriving a third stiffness matrix, which is the stiffness matrix of the specific shape portion, by subtracting a second stiffness matrix, which is the stiffness matrix of the first base portion, from the first stiffness matrix; and generating a second model in which shell elements, set as stiffness matrices in the third stiffness matrix, are attached to the second base portion.

This disclosure makes it possible to reduce the computational cost in finite element analysis.

Figure 1 is a block diagram showing an example of the functional configuration of an analysis device according to an embodiment of this disclosure. Figure 2 is a flowchart showing an example of the processing flow performed by the analysis device according to the embodiment of this disclosure. Figure 3 is a schematic diagram showing a model used in the processing performed by the analysis apparatus according to the embodiment of this disclosure, the model including a first base portion and a cusp portion. Figure 4 is a schematic diagram showing a model used in the processing performed by the analysis apparatus according to the embodiment of this disclosure, which is the model obtained by removing the cusp portion from the model in Figure 3. Figure 5 is a schematic diagram showing a model used in the processing performed by the analysis apparatus according to the embodiment of this disclosure, in which a shell element, whose third stiffness matrix is set as a stiffness matrix, is attached to the second base portion.

Embodiments of this disclosure will be described below with reference to the attached drawings. The dimensions, materials, and other specific numerical values shown in the embodiments are merely illustrative examples for ease of understanding and, unless otherwise specified, do not limit this disclosure. In this specification and the drawings, elements having substantially the same function or configuration are denoted by the same reference numerals to avoid redundant explanations, and elements not directly related to this disclosure are omitted from the illustrations.

Figure 1 is a block diagram showing an example of the functional configuration of the analysis device 1 according to this embodiment. The analysis device 1 performs finite element analysis to predict the stress and deformation behavior occurring in a component. The analysis device 1 includes, for example, a central processing unit (CPU), a ROM containing programs, and a RAM as a work area. As shown in Figure 1, the analysis device 1 includes, for example, a generation unit 11, an analysis unit 12, a specification unit 13, and a derivation unit 14. The functions of the analysis device 1 described below may be implemented by a single device or divided among multiple devices.

The generation unit 11 generates a model that reproduces the shape of the part to be analyzed using finite element analysis. For example, the generation unit 11 generates the model in response to user operations performed on the analysis device 1. The model used in finite element analysis is divided into multiple meshes (i.e., multiple elements). The generation of the meshes in the model is also performed by the generation unit 11.

The analysis unit 12 performs finite element analysis using the model generated by the generation unit 11. In the finite element analysis performed by the analysis unit 12, governing equations related to mechanics are set for each element of the model, and a solution that satisfies all of these governing equations is sought. Specifically, in the finite element analysis performed by the analysis unit 12, the governing equations set for each element are those that define the relationship between deformation and stress.

For example, equation (1) below is used as the governing equation set for each element. Equation (1) is an equation that defines the relationship between in-plane stress and in-plane strain, as well as the relationship between moment, curvature, and torsion. In the following explanation, it is assumed that the first and second directions are mutually orthogonal within the plane of the calculation target. Furthermore, the direction orthogonal to the first and second directions will be described as the third direction. In Figures 3 to 5, described later, the X and Y directions correspond to the first and second directions, respectively, and the Z direction corresponds to the third direction.

... (1)

In equation (1), _N11 , _N22 , and _N12 are in-plane stresses. _N11 is the normal stress in the first direction. _N22 is the normal stress in the second direction. _N12 is the shear stress in the plane parallel to the first and second directions. In equation (1), _M11 , _M22 , and _M12 are moments. _M11 is the bending moment about the axis in the first direction. _M22 is the bending moment about the axis in the second direction. _M12 is the torsional moment.

In equation (1), _ε₁¹ , _ε₂² , and _ε₁² are in-plane strains. _ε₁¹ is the normal strain in the first direction. _ε₂² is the normal strain in the second direction. _ε₁² is the shear strain in the plane parallel to the first and second directions. In equation (1), _κ₁¹ , _κ₂² , and _κ₁² are curvature and torsion ratios. _κ₁¹ is the curvature in the first direction. _κ₂² is the curvature in the second direction. _κ₁² is the torsion ratio.

In equation (1), K is the stiffness matrix that defines the relationship between in-plane stress and in-plane strain, as well as the relationship between moment, curvature, and torsion. The stiffness matrix K is expressed by the following equation (2).

... (2)

In equation (2), A is the in-plane stiffness matrix. In equation (2), B is the coupling stiffness matrix. In equation (2), D is the bending stiffness matrix. The in-plane stiffness matrix A defines the relationship between N ₁₁ , N ₂₂ , N ₁₂ and ε ₁₁ , ε ₂₂ , ε ₁₂ in equation (1). In other words, it defines the relationship between in-plane stress and in-plane strain. The bending stiffness matrix D defines the relationship between M ₁₁ , M ₂₂ , M ₁₂ and κ ₁₁ , κ ₂₂ , κ ₁₂ in equation (1). In other words, it defines the relationship between moment, curvature and torsion ratio.

For example, in addition to equation (1) above, equation (3) below is also used as a governing equation for each element. Equation (3) is an equation that defines the relationship between out-of-plane shear stress and out-of-plane shear strain.

... (3)

In equation (3), _S13 and _S23 are out-of-plane shear stresses. _S13 is the shear stress in the plane parallel to the first and third directions. _S23 is the shear stress in the plane parallel to the second and third directions. In equation (3), _γ13 and _γ23 are out-of-plane shear strains. _γ13 is the shear strain in the plane parallel to the first and third directions. _γ23 is the shear strain in the plane parallel to the second and third directions. In equation (3), E is the stiffness matrix that defines the relationship between out-of-plane shear stress and out-of-plane shear strain. The stiffness matrix E defines the relationship between _S13 , _S23 and _γ13 , _γ23 in equation (3). In other words, it defines the relationship between out-of-plane shear stress and out-of-plane shear strain.

The following describes an example in which equations (1) and (3) are used as governing equations in the finite element analysis performed by the analysis unit 12. However, as will be discussed later, the equations used as governing equations are not limited to this example.

The identification unit 13 identifies the stiffness matrix of the model used in the finite element analysis based on the analysis results of the finite element analysis. The derivation unit 14 uses the information obtained by the identification unit 13 to derive the stiffness matrix of a specific shape (for example, the cusp portion 31 in Figure 3, described later). In the analysis device 1, the computational cost of the finite element analysis is reduced primarily through the processing performed by the identification unit 13 and the derivation unit 14.

Figure 2 is a flowchart showing an example of the processing flow performed by the analysis device 1 according to this embodiment. The processing flow shown in Figure 2 is executed, for example, when a predetermined operation is performed by the user on the analysis device 1.

The following describes examples in which Model M1 (shown in Figure 3), Model M2 (shown in Figure 4), and Model M3 (shown in Figure 5) are used in the processing performed by the analysis device 1. However, the models used in the processing performed by the analysis device 1 are appropriately set according to the shape of the part being analyzed, and are not limited to these examples. As will be described later, Model M1 (shown in Figure 3) corresponds to an example of the first model. Model M2 (shown in Figure 4) corresponds to an example of the third model. Model M3 (shown in Figure 5) corresponds to an example of the second model.

When the processing flow shown in Figure 2 begins, in step S101, the generation unit 11 creates the model M1 shown in Figure 3. Model M1 is a model that reproduces a part of the component to be analyzed. As shown in Figure 3, model M1 includes a first base portion 21 and a cusp portion 31.

The cusp portion 31 corresponds to an example of a specific shape portion having a specific shape formed on the surface of the part being analyzed. The specific shape is a complex shape compared to a flat, simple shape. For example, the specific shape has at least a concave or convex portion. The cusp portion 31 has the shape of a machining mark formed on the workpiece. In the example in Figure 3, the cusp portion 31 is a portion where multiple concave portions extending in the X direction are arranged in the Y direction. The first base portion 21 is the portion of the part being analyzed that has the specific shape portion removed from the portion where the specific shape portion is formed. In other words, the first base portion 21 has a flat, simple shape. Specifically, the first base portion 21 has the basic shape of the workpiece. The specific shape portion has a more complex shape than the first base portion 21. In the example in Figure 3, the first base portion 21 has a flat plate shape. In Figure 3, the boundary between the first base portion 21 and the cusp portion 31 in model M1 is shown by a dashed line.

As shown in Figure 3, the model M1, viewed as a whole, is formed in a substantially flat plate shape, extending in the X and Y directions and having thickness in the Z direction. The first base portion 21 and the cusp portion 31 also extend in the X and Y directions, respectively, and have thickness in the Z direction. The first base portion 21 and the cusp portion 31 are adjacent to each other in the Z direction. Note that the X, Y, and Z directions are orthogonal to each other. The model M1 is divided into multiple elements. Specifically, as shown in Figure 3, the model M1 is divided in the X, Y, and Z directions.

Following step S101 in Figure 2, in step S102, the analysis unit 12 performs a finite element analysis using model M1. Specifically, the analysis unit 12 sets equations (1) and (3) for each element of model M1 and finds a solution that satisfies all the set governing equations. The stiffness matrices K and E set for each element are predetermined. Note that the stiffness matrices K and E may differ between elements in model M1.

Following step S102, in step S103, the identification unit 13 identifies the first stiffness matrix, which is the stiffness matrix of model M1. Specifically, the identification unit 13 identifies the first stiffness matrix based on the analysis results of a finite element analysis using model M1. The first stiffness matrix is the stiffness matrix K and stiffness matrix E of the entire model M1, assuming that the entire model M1 is a single element.

Following step S103, in step S104, the generation unit 11 creates the model M2 shown in Figure 4. As shown in Figure 4, model M2 is a model obtained by removing the cusp portion 31 from model M1 in Figure 3. In other words, model M2 is a model that reproduces only the first base portion 21, which has a flat and simple shape. As shown in Figure 4, model M2 is divided in the X, Y, and Z directions. In the example in Figure 4, the mesh cutting method of the first base portion 21 in model M2 is different from the mesh cutting method of the first base portion 21 in model M1 in Figure 3. However, the mesh cutting method of the first base portion 21 in model M2 may be the same as the mesh cutting method of the first base portion 21 in model M1.

Following step S104 in Figure 2, in step S105, the analysis unit 12 performs a finite element analysis using model M2. Specifically, the analysis unit 12 sets equations (1) and (3) for each element of model M2 and finds a solution that satisfies all the set governing equations. The stiffness matrices K and E set for each element are predetermined. Note that the stiffness matrices K and E may differ between elements in model M2.

Following step S105, in step S106, the identification unit 13 identifies the second stiffness matrix, which is the stiffness matrix of model M2. Since model M2 is a model that reproduces only the first base portion 21, the second stiffness matrix is the stiffness matrix of the first base portion 21. Specifically, the identification unit 13 identifies the second stiffness matrix based on the analysis results of a finite element analysis using model M2. The second stiffness matrix is the stiffness matrix K and stiffness matrix E of the entire first base portion 21, assuming the entire first base portion 21 as a single element.

Following step S106, in step S107, the derivation unit 14 derives the third stiffness matrix, which is the stiffness matrix of the cusp portion 31. Specifically, the derivation unit 14 derives the third stiffness matrix by subtracting the second stiffness matrix specified in step S106 from the first stiffness matrix specified in step S103. The third stiffness matrix is the stiffness matrix K and stiffness matrix E of the entire cusp portion 31, assuming the entire cusp portion 31 as a single element.

For example, the derivation unit 14 derives the stiffness matrix obtained by subtracting the stiffness matrix K of the first base portion 21, which was identified as the second stiffness matrix in step S106, from the stiffness matrix K of model M1, which was identified as the first stiffness matrix in step S103, as the stiffness matrix K of the cusp portion 31 among the third stiffness matrices. For example, the derivation unit 14 derives the stiffness matrix obtained by subtracting the stiffness matrix E of the first base portion 21, which was identified as the second stiffness matrix in step S106, from the stiffness matrix E of model M1, which was identified as the first stiffness matrix in step S103, as the stiffness matrix E of the cusp portion 31 among the third stiffness matrices.

Following step S107, in step S108, the generation unit 11 creates model M3 as shown in Figure 5. Model M3 is a model that reproduces a part of the component being analyzed that differs from the part reproduced by model M1. In reality, the surface of the part of the component being analyzed reproduced by model M3 has a cusp portion 31 formed thereon, similar to the part reproduced by model M1.

As shown in Figure 5, in Model M3, shell elements 41 are attached to the surface of the second base portion 22. Shell elements 41 are a type of element used in finite element analysis and are planar elements. Shell elements 41 are created on a surface. Although the shell elements 41 theoretically have rigidity equal to the plate thickness, they are not divided in the plate thickness direction. Specifically, multiple shell elements 41 are attached to the surface of the second base portion 22 so as to cover the entire surface area of the second base portion 22. Note that in Figure 5, for ease of understanding, the second base portion 22 and the shell elements 41 are shown separated. The second base portion 22 has a simple, flat shape, similar to the first base portion 21 in Model M1. In the example in Figure 5, the second base portion 22 has a flat plate shape. Note that since the parts reproduced by Model M1 and the parts reproduced by Model M3 are different, the shape and dimensions of the second base portion 22 are fundamentally different from those of the first base portion 21.

The shell element 41 is a shell element in which the third stiffness matrix derived in step S107 is set as the stiffness matrix. In other words, the shell element 41 is used in place of the solid element that reproduces the shape of the cusp portion 31. Multiple such shell elements 41 are attached to the surface of the second base portion 22 so as to cover the entire surface area of the second base portion 22. Therefore, model M3 is a model in which shell elements 41, in which the third stiffness matrix is set as the stiffness matrix, are attached to the second base portion 22.

Following step S108 in Figure 2, in step S109, the analysis unit 12 performs a finite element analysis using model M3, and the processing flow shown in Figure 2 is completed. Specifically, the analysis unit 12 sets equations (1) and (3) for each element of model M3, and finds a solution that satisfies all the set governing equations. The stiffness matrices K and E set for each element of the second base section 22 in model M3 are predetermined. The stiffness matrices K and E derived as the third stiffness matrices in step S107 are set for each shell element 41 in model M3.

As explained above, in the analysis device 1, the identification unit 13 identifies the first stiffness matrix, which is the stiffness matrix of the first model, based on the analysis results of a finite element analysis using a first model (model M1 in the above example) that includes a first base unit 21 and a specific shape part (cusp unit 31 in the above example). The derivation unit 14 derives the third stiffness matrix, which is the stiffness matrix of the specific shape part, by subtracting the second stiffness matrix, which is the stiffness matrix of the first base unit 21, from the first stiffness matrix. The generation unit 11 generates a second model (model M3 in the above example) by attaching shell elements 41, for which the third stiffness matrix is set as the stiffness matrix, to the second base unit 22. As a result, when performing finite element analysis on a part of the component to be analyzed in which a specific shape part is formed on the surface, finite element analysis can be performed without using a model that reproduces the shape of the specific shape part. In a model that reproduces the shape of the specific shape part (for example, model M1 above), the number of meshes becomes enormous as the complexity of the model's shape increases. On the other hand, the analysis device 1, by using shell elements 41, can perform finite element analysis without using a model that reproduces the shape of a specific part, thus suppressing the increase in the number of meshes that occurs with the complexity of the model's shape. Therefore, the computational cost in finite element analysis can be reduced.

Furthermore, the shell elements 41 generated by the processing flow shown in Figure 2 are used each time finite element analysis is performed on other parts of the component being analyzed where the cusp portion 31 is formed on the surface, after the completion of the processing flow shown in Figure 2. Specifically, in the finite element analysis of other parts of the component being analyzed where the cusp portion 31 is formed on the surface, the generation unit 11 generates a model by attaching the shell elements 41 generated by the above-described process to the base portion corresponding to that other part. Then, the analysis unit 12 performs finite element analysis using the obtained model. This allows for sequential finite element analysis of multiple parts of the component being analyzed where the cusp portion 31 is formed on the surface, while reducing computational costs.

In particular, the specific unit 13 identifies the second stiffness matrix based on the analysis results of a finite element analysis using a third model (model M2 in the above example) obtained by removing a specific shape portion (cusp portion 31 in the above example) from the first model (model M1 in the above example). This allows for the appropriate identification of the stiffness matrix of the entire first base portion 21 as the second stiffness matrix, assuming the entire first base portion 21 as a single element. However, the specific unit 13 may identify the second stiffness matrix without relying on the analysis results of a finite element analysis. For example, if the second stiffness matrix is known, the specific unit 13 can identify the second stiffness matrix without performing a finite element analysis. Since the calculation target for the second stiffness matrix is a simple plate shape, the second stiffness matrix may be obtained by manual calculation based on material mechanics, which is the mechanics of elastic bodies. For example, the specific unit 13 may identify the second stiffness matrix by calculations other than finite element analysis. In these cases, steps S104 and S105 may be omitted.

In particular, the first base portion 21 has the basic shape of the workpiece, while the specific shape portion (the cusp portion 31 in the above example) has a more complex shape than the first base portion 21. This reduces the computational cost in finite element analysis when the workpiece is the object of analysis. However, the object of analysis is not limited to the workpiece.

In particular, the specific shape is a cusp portion 31 having the shape of a machining mark formed on the workpiece. This allows for suppression of the increase in the number of meshes due to the complexity of the model's shape when performing finite element analysis on the portion of the part being analyzed in which the cusp portion 31 is formed on the surface. Therefore, the computational cost in finite element analysis of parts in which the cusp portion 31 is formed on the surface can be reduced. However, the specific shape is not limited to a cusp portion 31 and may have various shapes.

In particular, the stiffness matrix includes a matrix that defines the relationship between in-plane stress and in-plane strain (stiffness matrix K in equation (1) in the example above). This reduces the computational cost in finite element analysis for predicting in-plane stress and strain occurring in a component.

In particular, the stiffness matrix includes a matrix that defines the relationship between moment, curvature, and torsion (stiffness matrix K in equation (1) in the example above). This reduces the computational cost in finite element analysis for predicting the moment, curvature, and torsion occurring in a component.

In particular, the stiffness matrix includes a matrix that defines the relationship between out-of-plane shear stress and out-of-plane shear strain (stiffness matrix E in equation (3) in the example above). This reduces the computational cost in finite element analysis for predicting out-of-plane shear stress and out-of-plane shear strain occurring in a component.

While embodiments of this disclosure have been described above with reference to the attached drawings, it goes without saying that this disclosure is not limited to such embodiments. It will be obvious to those skilled in the art that various modifications or alterations can be conceived within the scope of the claims, and these will naturally fall within the technical scope of this disclosure.

The above describes an example in which equations (1) and (3) are used as governing equations in the finite element analysis performed by the analysis unit 12. However, the equations used as governing equations are not limited to the above example. For example, equation (1) alone may be used as the governing equation without using equation (3). For example, instead of equation (1), equations that define only the relationship between in-plane stresses _N11 , _N22 , _N12 and in-plane strains _ε11 , _ε22 , _ε12 may be used as the governing equation.

This disclosure can, for example, contribute to Sustainable Development Goal (SDG) 12, "Ensure sustainable consumption and production patterns."

1. Analysis device 11: Generation unit 13, Specification unit 14, Output unit 21, First base unit 22, Second base unit 31, Cusp unit (Specific shape unit)
41. Shell Element M1 Model (First Model)
M2 Model (Third Model)
M3 Model (2nd Model)

Claims (8)

A specific unit identifies a first stiffness matrix, which is the stiffness matrix of the first model, based on the analysis results of a finite element analysis using a first model including a first base unit and a specific shape unit.
A derivation unit that derives a third stiffness matrix, which is the stiffness matrix of the specific shape portion, by subtracting the second stiffness matrix, which is the stiffness matrix of the first base portion, from the first stiffness matrix,
A generation unit that generates a second model in which the shell elements, set as the third stiffness matrix, are attached to the second base portion,
Equipped with,
Analysis device. The specified part determines the second stiffness matrix based on the analysis results of a finite element analysis using a third model obtained by removing the specified shape part from the first model.
The analysis apparatus according to claim 1. The first base portion has the basic shape of the workpiece,
The aforementioned specific shaped portion has a more complex shape than the first base portion.
The analysis apparatus according to claim 1 or 2. The aforementioned specific shaped portion has the shape of a processing mark formed on the workpiece.
The analysis apparatus according to claim 3. The stiffness matrix includes a matrix that defines the relationship between in-plane stress and in-plane strain.
The analysis apparatus according to claim 1 or 2. The stiffness matrix includes a matrix that defines the relationship between moment, curvature, and torsion.
The analysis apparatus according to claim 1 or 2. The stiffness matrix includes a matrix that defines the relationship between out-of-plane shear stress and out-of-plane shear strain.
The analysis apparatus according to claim 1 or 2. A step of identifying a first stiffness matrix, which is the stiffness matrix of the first model, based on the analysis results of a finite element analysis using a first model including a first base portion and a specific shape portion.
The steps include: subtracting the second stiffness matrix, which is the stiffness matrix of the first base portion, from the first stiffness matrix to derive the third stiffness matrix, which is the stiffness matrix of the specific shaped portion;
The steps include generating a second model by attaching shell elements, for which the third stiffness matrix is set as a stiffness matrix, to the second base portion,
including,
Analysis method.