Fluid-solid-sound coupling calculation method for time domain analysis based on OpenFOAM platform

Abstract

The invention discloses a fluid-solid-sound coupling calculation method based on time domain analysis of an OpenFOAM platform. The fluid-solid-sound coupling calculation method comprises the following steps: decomposing a compressible fluid variable into an incompressible fluid variable and an acoustic pulsation quantity; deducing an acoustic control equation of a flow sound decomposition method under the dynamic grid; discretizing the acoustic control equation by adopting a finite volume method; applying an incompressible flow field boundary condition and an acoustic boundary condition; adopting an OpenFOAM dynamic grid technology to obtain grid information topological information and motion information; a PISO-SIMPLE merging algorithm is adopted to decouple the acoustic control equation; and solving the decoupled acoustic control equation to obtain sound pressure distribution so as to realize fluid-structure interaction vibration noise characteristic analysis. According to the method, the technologies of variable decomposition, fluid-solid coupling and dynamic grids are utilized, far-field linear hypothesis is avoided, acoustic forecasting of far and near fields is guaranteed, meanwhile, the calculation cost is saved, and the analysis efficiency is improved.

Description

Technical Field

[0001] The present invention belongs to the technical field of fluid-solid-acoustic coupling and acoustic radiation, and in particular relates to a fluid-solid-acoustic coupling calculation method based on time domain analysis of an OpenFOAM platform. Background Art

[0002] Reducing flow noise is a key task in the aerospace, shipbuilding and automotive industries. In the development of a new generation of quiet vehicles, there is a widespread fluid-solid-acoustic coupling effect, that is, the interaction between fluid and structure, which affects the generation and propagation of sound. The study of the underwater fluid-solid-acoustic coupling mechanism naturally becomes one of the key basic issues in the development of submarine acoustic stealth technology. For example, the composite propeller developed in recent decades has the characteristics of good noise performance compared with traditional metal propellers, which can reduce the flow noise of submarine propellers. At the same time, the blades of composite propellers will produce large deformations during operation, and the influence of structural deformation on the surrounding flow field and sound field must be considered during design. In recent years, most scholars have focused on the problem of acoustic radiation, usually focusing on free shear flow and wall shear flow under stationary grids. However, in actual situations, the deformation and aeroacoustic behavior of elastic bodies are heavily dependent on the coupling between fluid and solid. Like other scientific fields, there are three most basic methods for studying fluid-solid-acoustic coupling problems, namely experimental method, analytical method and numerical simulation method. Most fluid-solid-acoustic coupling problems are difficult to obtain analytical solutions, and experimental methods are limited by scale effects and high costs. Numerical simulation methods have the inherent advantages of short cycle, low cost and no environmental influence, which is of great significance for the study of complex fluid-solid-acoustic coupling problems. Summary of the invention

[0003] In order to solve the above technical problems, the present invention proposes a fluid-solid-acoustic coupling calculation method based on the time domain analysis of the OpenFOAM platform, which supports multi-core parallel computing, has a user-friendly partial differential equation description syntax and a powerful open source post-processing platform. The present invention adopts a hybrid prediction method to deconstruct the sound source and can accurately and quickly predict and analyze the sound radiation.

[0004] To achieve the above object, the present invention provides a fluid-solid-acoustic coupling calculation method based on the time domain analysis of the OpenFOAM platform, comprising:

[0005] Decomposing the compressible fluid variables into incompressible fluid variables and acoustic pulsation quantities;

[0006] Derive the acoustic governing equations of the flow-acoustic decomposition method under a moving grid;

[0007] The finite volume method is used to discretize the acoustic control equations;

[0008] Apply incompressible flow boundary conditions and acoustic boundary conditions;

[0009] Use OpenFOAM dynamic mesh technology to obtain mesh topology and motion information;

[0010] The PISO-SIMPLE combined algorithm is used to decouple the acoustic control equations;

[0011] The decoupled acoustic control equations are solved to obtain the sound pressure distribution and realize the analysis of fluid-solid coupling vibration and noise characteristics.

[0012] Optionally, the compressible fluid variables are decomposed into incompressible fluid variables and acoustic pulsation quantities as:

[0013] ρ=ρ0+ρ a

[0014] u=U+u a

[0015] p=P+p a

[0016] Where ρ, u, and p are the density, velocity, and pressure of the compressible fluid, respectively; ρ0, U, and P are the density, velocity, and pressure of the incompressible fluid, respectively; ρ a 、u a 、p a are the acoustic pulsation density, pressure and velocity, respectively.

[0017] Optionally, the acoustic governing equations of the flow-acoustic decomposition method under a moving grid are derived as:

[0018]

[0019] Among them, u g is the grid velocity, f is the acoustic intermediate vector, and T is the viscosity term of the acoustic equation.

[0020] Optionally, the finite volume method is used to discretize the governing acoustic equations including:

[0021] Based on the incompressible NS equations and the acoustic control equations, the time term is discretized using the Euler or Backward format;

[0022] Based on the incompressible NS equations and the acoustic control equations, the TVD or NVD format is used for the convection term, the modified format is used for the diffusion term, and the Gaussian integral transform is used for the gradient term and the source term, and the spatial terms are constructed after merging.

[0023] Optionally, apply incompressible flow boundary conditions and acoustic boundary conditions including:

[0024] Setting incompressible flow field boundary conditions, the incompressible flow field boundary conditions include velocity inlet, pressure outlet and solid wall no-slip, wherein the velocity inlet refers to the value of the velocity at the specified inlet, the pressure outlet refers to the directional derivative of the pressure at the outlet is specified as a fixed value, the solid wall no-slip refers to the value of the wall velocity is specified as a corresponding fixed value, and the directional derivative of the wall pressure is specified as zero;

[0025] An acoustic boundary condition is set, wherein the acoustic boundary condition is divided into a solid wall boundary condition and a non-reflecting boundary condition.

[0026] Optionally, OpenFOAM dynamic mesh technology is used to obtain mesh information topology information and motion information including:

[0027] For rigid bodies, the dynamicMotionSolverFvMesh provided by OpenFOAM is used to solve the deformation of fluid-solid coupling and realize the motion of rigid bodies with six degrees of freedom;

[0028] For deformable bodies, the unsIncrTotalLagrangianSolid and unsTotalLagrangianSolid solid solvers provided by OpenFOAM are combined to obtain the mesh topology information and motion information by solving the Laplacian equation at the center of the body.

[0029] Optionally, the PISO-SIMPLE combined algorithm is used to decouple the acoustic control equations including:

[0030] Solve the momentum equation in the acoustic control equation, calculate and update the acoustic intermediate vector f;

[0031] Using the preliminary solution to the momentum equation, solve the continuity equation of the acoustic governing equation and update the acoustic pulsation pressure p a ;

[0032] According to the corrected acoustic pulsation pressure p a , further correct the acoustic intermediate vector f;

[0033] On this basis, the acoustic pulsation pressure p a The correction and acoustic intermediate vector f correction steps are combined with the overall multiple iterations of the continuity equation until the calculation of the current time step is completed, or the overall result meets the preset convergence accuracy requirements, the calculation is terminated.

[0034] Technical effect of the present invention: The present invention discloses a fluid-solid-acoustic coupling calculation method for time-domain analysis based on the OpenFOAM platform, which decouples the entire calculation into incompressible flow field calculation, dynamic grid calculation and acoustic calculation. The difficulty of the entire calculation is greatly reduced, and the efficiency and accuracy are improved. After considering the influence of the interaction between fluid and solid, this will also be beneficial for a deep understanding of the mechanism of acoustic radiation. Specifically, the acoustic control equation is strictly derived from the compressible fluid equation and the incompressible fluid equation, and the far field is not assumed to be linear propagation, avoiding the applicability problems caused by the acoustic assumption. In addition, based on the OpenFOAM platform, the calculation of fluid-solid-acoustic adopts the finite volume method, which ensures the uniformity of the algorithm, avoids the use of the finite volume method for fluid dynamics, the finite element method for structural mechanics and acoustics, and the errors caused by the information mapping of different methods, greatly saves the calculation time, and improves the calculation efficiency of the analysis process. Finally, the program of the present invention has a high degree of flexibility. With the growth of the OpenFOAM community and the enrichment of algorithms and physical models in OpenFOAM, the program is easy to expand, which greatly reduces the cost of development. BRIEF DESCRIPTION OF THE DRAWINGS

[0035] The drawings constituting a part of the present application are used to provide a further understanding of the present application. The illustrative embodiments and descriptions of the present application are used to explain the present application and do not constitute an improper limitation on the present application. In the drawings:

[0036] Figure 1 It is a flow chart of a fluid-solid-acoustic coupling calculation method based on time domain analysis of the OpenFOAM platform according to an embodiment of the present invention;

[0037] Figure 2 Schematic diagram of the process of the PISO-SIMPLE merging algorithm according to an embodiment of the present invention;

[0038] Figure 3 A schematic diagram of interpolation on a non-orthogonal grid according to an embodiment of the present invention;

[0039] Figure 4 A schematic diagram of parameters in the non-reflection boundary equation of an embodiment of the present invention;

[0040] Figure 5 Schematic diagram of the calculation domain of flow around an oscillating cylinder according to an embodiment of the present invention;

[0041] Figure 6 A schematic diagram of force distribution at different reduced speeds according to an embodiment of the present invention;

[0042] Figure 7 Schematic diagram of amplitude distribution at different reduced speeds according to an embodiment of the present invention;

[0043] Figure 8Schematic diagram of lateral amplitude distribution at different reduced speeds according to an embodiment of the present invention;

[0044] Fig. 9 Schematic diagram of the instantaneous sound field cloud of flow around an oscillating cylinder according to an embodiment of the present invention. DETAILED DESCRIPTION

[0045] It should be noted that, in the absence of conflict, the embodiments and features in the embodiments of the present application can be combined with each other. The present application will be described in detail below with reference to the accompanying drawings and in combination with the embodiments.

[0046] It should be noted that the steps shown in the flowcharts of the accompanying drawings can be executed in a computer system such as a set of computer executable instructions, and that, although a logical order is shown in the flowcharts, in some cases, the steps shown or described can be executed in an order different from that shown here.

[0047] like Figure 1 As shown, this embodiment provides a fluid-solid-acoustic coupling calculation method based on the time domain analysis of the OpenFOAM platform, including:

[0048] Decomposing the compressible fluid variables into incompressible fluid variables and acoustic pulsation quantities;

[0049] Derive the acoustic governing equations of the flow-acoustic decomposition method under a moving grid;

[0050] The finite volume method is used to discretize the acoustic control equations;

[0051] Apply incompressible flow boundary conditions and acoustic boundary conditions;

[0052] Use OpenFOAM dynamic mesh technology to obtain mesh topology and motion information;

[0053] The PISO-SIMPLE combined algorithm is used to decouple the acoustic control equations;

[0054] The decoupled acoustic control equations are solved to obtain the sound pressure distribution and realize the analysis of fluid-solid coupling vibration and noise characteristics.

[0055] Furthermore, the compressible fluid variables are decomposed into incompressible fluid variables and acoustic pulsation variables as follows:

[0056] ρ=ρ0+ρ a

[0057] u=U+u a

[0058] p=P+p a

[0059] Where ρ, u, and p are the density, velocity, and pressure of the compressible fluid, respectively; ρ0, U, and P are the density, velocity, and pressure of the incompressible fluid, respectively; ρ a 、u a 、p a are the acoustic pulsation density, pressure and velocity, respectively.

[0060] Furthermore, the acoustic control equation of the flow-acoustic decomposition method under the moving grid is derived as follows:

[0061]

[0062] Among them, u g is the grid velocity, f is the acoustic intermediate vector, and T is the viscosity term of the acoustic equation.

[0063] Furthermore, the finite volume method is used to discretize the acoustic control equations, including:

[0064] Based on the incompressible NS equations and the acoustic control equations, the time term is discretized using the Euler or Backward format;

[0065] Based on the incompressible NS equations and the acoustic control equations, the TVD or NVD format is used for the convection term, the modified format is used for the diffusion term, and the Gaussian integral transform is used for the gradient term and the source term, and the spatial terms are constructed after merging.

[0066] Specifically, there are two forms of expression for time term discreteness:

[0067] Method 1:

[0068]

[0069] Method 2:

[0070]

[0071] Among them, method 1 is the Euler format with first-order accuracy, method 2 is the backward format with second-order accuracy, φ is a discrete variable, p is the control volume, n represents the current moment, n-1 represents the previous moment, and n-2 represents the previous moment;

[0072] The spatial discretization mainly involves the discretization of convection terms, diffusion terms, gradient terms and source terms.

[0073] The convection term ∫ V ▽·(ΨΦ) as an example, based on Gauss's theorem, the volume integral is converted into the surface integral, and we get

[0074]

[0075] Wherein, the subscript f is the quantity on the grid surface, which can be obtained by interpolating the quantity at the center point of the grid, F = Φ f ·S f is the flux;

[0076] There are two forms of discrete expressions for convection terms:

[0077] Method 1:

[0078] Φ f =Φ1+Ψ(r)(Φ HO -Φ1)

[0079]

[0080] Method 2:

[0081]

[0082] Among them, method 1 is TVD format, and method 2 is NVD format. Compared with TVD format, the restrictions are loose. Φ1 represents the interpolation obtained by the first-order format, Φ HO represents the interpolation obtained using the second-order central format, P, N are the grid center points adjacent to the interface f, and the flow direction is from P to N;

[0083] The discretization of the diffusion term can be expressed as:

[0084]

[0085] Among them, Δ nc =1 / cosθ, the schematic diagram of interpolation on non-orthogonal grid is as shown in the attached figure. Figure 3 As shown;

[0086] Based on Gauss's theorem, the method of converting volume integral into surface integral is adopted. The specific expression is as follows:

[0087]

[0088] The equations that cannot be written in the above form can be attributed to the source term. The source term can appear in the control equation in the form of an expression without discretization.

[0089] Furthermore, the incompressible flow field boundary conditions and acoustic boundary conditions are applied including:

[0090] The boundary conditions of the incompressible flow field are set as velocity inlet, pressure outlet and solid wall no-slip, wherein the velocity inlet specifies the value of the velocity at the inlet, the pressure outlet specifies the directional derivative of the pressure at the outlet as 0, and the solid wall no-slip specifies the value of the wall velocity as 0 and the directional derivative of the wall pressure as 0;

[0091] The acoustic boundary conditions are divided into solid wall boundary conditions and non-reflecting boundaries.

[0092] Specifically, the acoustic solid wall boundary condition is handled in the same way as the incompressible fluid boundary condition, except that the variable to be determined is the acoustic pulsation quantity. The following characteristic non-reflecting boundary proposed by Tam is adopted, which refers to the fact that the acoustic boundary condition is explicitly updated, that is, the initial state of the boundary value is calculated based on the previous moment, and is continuously modified in the cycle according to the change of the value of the adjacent unit of the boundary;

[0093]

[0094] Where V g =(c+U+u a )·e r , e r The unit direction vector from the boundary to the sound source, r is the distance from the boundary to the sound source, such as Figure 4 As shown, for two-dimensional problems n=0.5, and for three-dimensional problems n=1.

[0095] Furthermore, OpenFOAM dynamic mesh technology is used to obtain mesh information topology information and motion information including:

[0096] For rigid bodies, the dynamicMotionSolverFvMesh provided by OpenFOAM is used to solve the deformation of fluid-solid coupling, and the motion of the rigid body is realized with six degrees of freedom. The motion control equation is as follows:

[0097]

[0098] Among them, m is the mass of the cylinder, c is the structural damping coefficient, k is the structural stiffness coefficient, and F is the resistance and lift of the cylinder respectively;

[0099] For deformable bodies, the unsIncrTotalLagrangianSolid and unsTotalLagrangianSolid solid solvers provided by OpenFOAM are combined to obtain the mesh topology information and motion information by solving the Laplacian equation at the center of the body.

[0100] Furthermore, the PISO-SIMPLE combined algorithm is used to decouple the acoustic control equations, including:

[0101] Specifically, for the incompressible NS equations, we can use Figure 2 The PISO-SIMPLE algorithm shown in the figure is used for decoupling; because the acoustic control equations are similar in form to the incompressible NS equations, the same PISO-SIMPLE algorithm is used for the structure of the acoustic control equations; as shown in the figure Figure 3As shown, the PISO-SIMPLE algorithm is used to decouple the momentum equation and continuity equation of the incompressible fluid through three steps: First, solve the momentum equation in the incompressible control equation, and calculate and update the incompressible fluid velocity U. Use the preliminary solution of the momentum equation to solve the continuity equation of the incompressible control equation, thereby updating the pressure P. According to the corrected pressure P, the incompressible fluid velocity U is further corrected. On this basis, the pressure P correction and velocity U correction steps are repeated, combined with multiple iterations of the continuity equation. The calculation is terminated until the calculation of the current time step is completed, or the overall result meets the preset convergence accuracy requirements. It is worth noting that the decoupling solution process of the acoustic equation can only be carried out under the premise that the incompressible flow field equation is decoupled and solved and converged at each time step.

[0102] Furthermore, solving the decoupled acoustic control equation to obtain the fluid-solid coupling vibration noise characteristics includes: first, solving the momentum equation in the acoustic control equation, calculating and updating the acoustic intermediate vector f. Using the preliminary solution of the momentum equation, solving the continuity equation of the acoustic control equation, and thus updating the acoustic pulsation pressure p a According to the corrected acoustic pulsation pressure p a , further correct the acoustic intermediate vector f. On this basis, the acoustic pulsation pressure p is repeatedly a The correction and acoustic intermediate vector f correction steps are combined with the overall multiple iterations of the continuity equation until the calculation of the current time step is completed or the overall result meets the preset convergence accuracy requirements, and then the calculation ends.

[0103] Specifically, by calculating the flow field equations and rigid body motion or flexible deformation, the incompressible flow field and grid information can be obtained. Then, by solving the acoustic control equations, the acoustic pulsation velocity, pressure and density can be calculated. The acoustic pressure has the following expression: By obtaining the sound pressure distribution, the fluid-solid coupling vibration and noise characteristics can be analyzed.

[0104] In order to verify the technical effect, the present invention considers the fluid-solid coupling and acoustic propagation characteristics of a cylinder doing sinusoidal motion in a fluid with a mass ratio m*=3 and a reduced velocity Ur=4. Figure 5 As shown in the figure, when calculating the flow field, the left half of the circle is the velocity inlet, the right half of the circle is the pressure outlet, and the cylinder is non-slip. In the acoustic field calculation, both sides are non-reflecting boundary conditions, and the cylinder is set as a solid wall boundary condition. The cylinder is forced to vibrate in the flow field with an amplitude of 0.2D, and the frequency of the fixed cylinder vibration is 15.97Hz. The time step of the calculation is Δt=0.001s, and the total calculation time is 10s. When t=3s, the flow field calculation area changes periodically, and the acoustic field calculation begins. Figures 6 to 8The obtained curves of the force coefficient, amplitude and oscillation frequency of the flow field of the oscillating cylinder as a function of the reduced velocity show good agreement with the references. Fig. 9 As shown, the instantaneous sound pressure field of the oscillating cylinder obtained by the present invention has no far-field linear assumption, and can realize the visualization of near-field and long-range sound pressure. By adopting the method of flow-acoustic separation and a larger time step, it is not difficult to find that the present invention reduces the amount of calculation to a great extent. In addition, it is found from the cloud map that the sound field is characterized by a dipole distribution, and the sound waves propagate with different intensities throughout the space. Unlike the sound field of a stationary cylinder, the "jumping" phenomenon of this sound field can only be found when considering structural vibration, which further verifies the importance of considering the multi-physics field of fluid-solid-acoustic coupling for exploring the essence of the sound field problem.

[0105] The present invention is an extension of the flow-acoustic decomposition method, which makes the analysis model more suitable for the actual model. The method of variable decomposition is used to solve the compressible NS equations in a decoupled manner, without citing the far-field linear assumption. The incompressible flow field variables are used as the input of the sound field. By solving the derived acoustic perturbation equations of flow-solid-acoustic under the dynamic grid, the calculation of the far and near fields can be realized at the same time, and the physical mechanism hidden behind the device can be understood more clearly. The open source software OpenFOAM is used for secondary development, and the finite volume method is used in the same platform to simulate the multi-physical field of flow-solid-acoustic, which can avoid the mapping error caused by the information transmission between different grids, greatly save the calculation time, and improve the calculation efficiency of the analysis process.

[0106] The above are only preferred specific implementations of the present application, but the protection scope of the present application is not limited thereto. Any changes or substitutions that can be easily thought of by a person skilled in the art within the technical scope disclosed in the present application should be included in the protection scope of the present application. Therefore, the protection scope of the present application should be based on the protection scope of the claims.

Claims

1. The fluid-solid-acoustic coupling calculation method based on the time domain analysis of the OpenFOAM platform is characterized by: include: Decomposing the compressible fluid variables into incompressible fluid variables and acoustic pulsation quantities; Derive the acoustic governing equations of the flow-acoustic decomposition method under a moving grid; The finite volume method is used to discretize the acoustic control equations; Apply incompressible flow boundary conditions and acoustic boundary conditions; Use OpenFOAM dynamic mesh technology to obtain mesh topology and motion information; The PISO-SIMPLE combined algorithm is used to decouple the acoustic control equations; The decoupled acoustic control equations are solved to obtain the sound pressure distribution and realize the analysis of fluid-solid coupling vibration and noise characteristics.

2. The fluid-solid-acoustic coupling calculation method based on the time domain analysis of the OpenFOAM platform according to claim 1, characterized in that: Decomposing the compressible fluid variables into incompressible fluid variables and acoustic pulsation variables is: p=p0+p a u=U+u a p=P+p a Where ρ, u, and p are the density, velocity, and pressure of the compressible fluid, respectively; ρ0, U, and P are the density, velocity, and pressure of the incompressible fluid, respectively; ρ a 、u a 、p a are the acoustic pulsation density, pressure and velocity, respectively.

3. The fluid-solid-acoustic coupling calculation method based on the time domain analysis of the OpenFOAM platform according to claim 1, characterized in that: The acoustic momentum equation and continuity equation control equations of the flow-acoustic decomposition method under the moving grid are derived as follows: Among them, u g is the grid velocity, f is the acoustic intermediate vector, and T is the viscosity term of the acoustic equation.

4. The fluid-solid-acoustic coupling calculation method based on the time domain analysis of the OpenFOAM platform according to claim 1, characterized in that: The finite volume method is used to discretize the acoustic control equations including: Based on the incompressible NS equations and the acoustic control equations, the time term is discretized using the Euler or Backward format; Based on the incompressible NS equations and the acoustic control equations, the TVD or NVD format is used for the convection term, the modified format is used for the diffusion term, and the Gaussian integral transform is used for the gradient term and the source term, and the spatial terms are constructed after merging.

5. The fluid-solid-acoustic coupling calculation method based on the time domain analysis of the OpenFOAM platform according to claim 1, characterized in that: Applying incompressible flow boundary conditions and acoustic boundary conditions includes: Setting incompressible flow field boundary conditions, the incompressible flow field boundary conditions include velocity inlet, pressure outlet and solid wall no-slip, wherein the velocity inlet refers to the value of the velocity at the specified inlet, the pressure outlet refers to the directional derivative of the pressure at the outlet is specified as a fixed value, the solid wall no-slip refers to the value of the wall velocity is specified as a corresponding fixed value, and the directional derivative of the pressure at the wall is specified as zero; An acoustic boundary condition is set, wherein the acoustic boundary condition is divided into a solid wall boundary condition and a non-reflecting boundary condition.

6. The fluid-solid-acoustic coupling calculation method based on the time domain analysis of the OpenFOAM platform according to claim 1, characterized in that: Using OpenFOAM dynamic mesh technology, the mesh information topology information and motion information are obtained, including: For rigid bodies, the dynamicMotionSolverFvMesh provided by OpenFOAM is used to solve the deformation of fluid-solid coupling and realize the motion of rigid bodies with six degrees of freedom; For deformable bodies, the unsIncrTotalLagrangianSolid and unsTotalLagrangianSolid solid solvers provided by OpenFOAM are combined to obtain the mesh topology information and motion information by solving the Laplacian equation at the center of the body.

7. The fluid-solid-acoustic coupling calculation method based on the time domain analysis of the OpenFOAM platform according to claim 1, characterized in that: The decoupling of the acoustic control equations by using the PISO-SIMPLE combined algorithm includes: solving the momentum equation in the acoustic control equations, calculating and updating the acoustic intermediate vector f; Using the preliminary solution to the momentum equation, solve the continuity equation of the acoustic governing equation and update the acoustic pulsation pressure p a ; According to the corrected acoustic pulsation pressure p a , further correct the acoustic intermediate vector f; On this basis, the acoustic pulsation pressure p a The correction and acoustic intermediate vector f correction steps are combined with the overall multiple iterations of the continuity equation until the calculation of the current time step is completed, or the overall result meets the preset convergence accuracy requirements, the calculation is terminated.

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