A fast fluid-structure interaction simulation method for heart valves based on isogeometric analysis

Abstract

The present invention relates to a rapid simulation method for fluid-solid coupling of heart valves based on isogeometric analysis, comprising step 1: optimizing fluid subproblems: simplifying the background grid of the fluid subproblems, generating Bézier tetrahedron background grids based on the simplified grids, and solving the fluid subproblems using the geometrically irrelevant field approximation idea; step 2: optimizing solid subproblems: using a strategy to divide the NURBS surface used in the solid subproblem into sliced ​​NURBS surfaces; step 3: using an augmented dynamic Lagrange multiplier method to couple the fluid-solid subproblems; step 4: solving the fluid-solid coupling problem and analyzing the results, and using the method proposed in this article to optimize the problem during the computer simulation of the heart valve, the efficiency of the simulation program can be improved while ensuring a certain accuracy. Compared with relevant research in recent years, the heart valve optimization simulation algorithm in the present invention is about 49% faster.

Description

Technical Field

[0001] The present invention relates to a fast simulation method for fluid-structure interaction of heart valves based on isogeometric analysis, belonging to the field of fluid-structure interaction simulation analysis. Background Technique

[0002] The incidence of cardiovascular diseases has been increasing year by year, and the research on cardiac physiology and pathology has received extensive attention from domestic and foreign experts and scholars. Using a computer to simulate heart valves is often used as an important method for studying various physiological mechanisms of the heart because it is not restricted by medical ethics and social ethics and has strong repeatability.

[0003] In fact, simulating heart valve problems requires solving a problem of coupling the Navier-Stokes equations (NS equations) in fluid mechanics and equations in structural mechanics. This problem has high mathematical and geometric complexity and is restricted by computing resources. Currently, certain compromises have been made in the modeling complexity of heart valve simulations. However, when using existing algorithms and computing resources for heart valve simulation, a large amount of time cost is still required. Therefore, it is very meaningful to optimize the fluid-structure interaction framework for solving heart valve problems.

[0004] Currently, there is an analysis method for optimizing the mechanical properties of artificial heart valves based on ANSYS / Workbench (publication number CN104758093B). The above method is based on the hydrodynamics of heart valves, clarifies the theory of the interaction between the fluid domain (blood) and the solid domain (artificial heart valve), and conducts fluid-structure interaction analysis of biological valves by constructing a fluid-structure interaction finite element model of biological valves. Taking the configuration of the human natural heart valve as the prototype and combining the reference surface of the valve leaf in the current research field, four rotational surfaces, namely, a spherical surface, a cylindrical surface, a paraboloid of revolution surface, and an ellipsoidal surface, are selected as the reference surfaces of the biological valve prototype for analysis. Summary of the Invention

[0005] The purpose of the present invention is to provide a rapid simulation method for heart valve fluid-solid coupling based on isogeometric analysis in view of the limitations and shortcomings of the prior art. For the fluid sub-problem, the fluid background grid is first simplified, and then the simplified grid is used to solve the fluid velocity field and pressure. In order to maintain accuracy, the present invention uses the idea of ​​geometrically independent field approximation (GIFT); in the structural mechanics simulation of the valve leaflet, the present invention uses piecewise spline surfaces to represent the valve leaflet geometric model, and in order not to introduce singular points, a "copper coin" structure is proposed for piecewise representation. In order to obtain higher simulation accuracy, the present invention adopts an immersed geometry strategy in solving the deformation problem of the heart valve leaflet; in the coupled fluid-solid control equation, a dynamically augmented Lagrangian algorithm is used; and in time discretization, a generalized-α method is used to control high-frequency dissipation.

[0006] To achieve the above object, the technical solution of the present invention is:

[0007] A fast simulation method for fluid-structure coupling of heart valves based on isogeometric analysis includes the following steps:

[0008] Step 1: Optimize the fluid subproblem: simplify the background mesh of the fluid subproblem, generate a Bézier tetrahedron background mesh based on the simplified mesh, and solve the fluid subproblem using the geometrically independent field approximation (GIFT) idea;

[0009] Step 2: Optimize the solid subproblem: Use a certain strategy to split the NURBS surface used in the solid subproblem into sliced ​​NURBS surfaces;

[0010] Step 3: Use the augmented dynamic Lagrange multiplier method to couple the fluid-solid problem;

[0011] Step 4: Solve the fluid-structure interaction problem and analyze the results.

[0012] Preferably, in the steps 1 to 2, in order to solve the loss of geometric accuracy after simplification, the present invention generates a Bézier tetrahedron mesh based on the simplified tetrahedron mesh. Because the Bézier tetrahedron belongs to a high-order mesh, according to the idea of ​​isoparametric elements, a higher-order mesh can obtain higher accuracy with relatively fewer degrees of freedom. In this paper, the Bézier tetrahedron mesh is used as the background mesh in the solution of the fluid subproblem.

[0013] Preferably, in order to minimize the difference between the result solved by the present invention and the solution accuracy using the un-simplified mesh as much as possible, it is not sufficient to represent the geometry only with high-order meshes because the generated Bézier meshes have fewer degrees of freedom. To obtain higher accuracy, higher-order basis functions should be used in the process of physical field approximation. According to the paradigm of isogeometric analysis, the basis functions used in the analysis process are the same as those used to represent the geometry. Although this can accurately represent the geometry in the analysis process, it limits the flexibility of isogeometric analysis. If higher-order basis functions are to be used in the analysis, the geometry needs to be modified and upgraded. Atroshchenko et al. proposed a new discretization scheme called the Generalized Isogeometric Analysis with Different Spline Spaces for the Computational Domain and Physical Fields (GIFT). This method can independently select the spline basis function spaces for geometry and the analysis process. Given a computational domain in spline form, different spline representations can be used in the analysis process using this method. The generalized isogeometric analysis based on this idea has the following characteristics:

[0014] One can, while maintaining the geometric accuracy of the domain boundary, select different spline spaces with different properties according to the form of the solution of the problem, such as the continuity of the solution, boundary layers, and singularities;

[0015] Two, for multi-patch analysis where the computational domain consists of multiple spline patches, the continuity conditions between adjacent patches can be well preserved without being restricted by the variable form;

[0016] Three, refinement operations such as knot insertion and degree elevation to increase degrees of freedom can be independent of the spline space of the computational domain and be directly performed in the spline space of the approximate solution.

[0017] The present invention borrows the GIFT idea to use higher-order polynomial basis functions in the simulation process without modifying the geometry, thereby accelerating the simulation solution speed while ensuring a certain accuracy.

[0018] Preferably, in step 2-1, in order to overcome the defect of NURBS surface degeneration at the boundary, the NURBS surface of each valve leaflet is represented by patches in the present invention. In addition, in order not to generate singular points after patching, a structure similar to a "copper coin" is used in the patching strategy to divide the surface. The geometric model of the valve leaflet is represented by patches through the "copper coin" structure template. Solving the strain using the patched valve leaflet can effectively accelerate the simulation speed and achieve higher accuracy at the boundary, that is, obtain a more accurate solution at the boundary. Since the geometric model of the patched valve leaflet does not degenerate at the boundary, that is, it can represent the same valve leaflet model geometrically as before without using a large number of control points at the boundary, it has fewer degrees of freedom. However, solving the deformation problem (solid sub-problem) of the patched valve leaflet geometry requires some modifications to the existing program and the application of some constraints.

[0019] Preferably, in step three, in order to obtain higher accuracy and overcome the problem of sudden change of the velocity field during the cardiac contraction phase, the present invention uses the dynamic augmented Lagrangian method to couple the solid and fluid equations. This method is a two-way coupling method and is essentially an improvement of the Lagrange multiplier method;

[0020] Compared with the prior art, the beneficial effects of the present invention are as follows:

[0021] Regarding the fluid-structure interaction rapid simulation method of heart valves based on isogeometric analysis involved in the present invention, for the fluid-structure interaction simulation of heart valves, most of the previous simulation methods take a lot of time. In the present invention, we propose an improved simulation method to accelerate the simulation process. For the fluid sub-problem, high-order Bézier tetrahedrons are used instead of linear tetrahedral meshes to represent the geometric model of the heart lumen. Using the idea of GIFT, basis functions different from geometric representation are used to approximate the physical field. For the solid sub-problem, multi-patch NURBS surfaces are used to represent the geometry, overcoming the problem of geometric patch degeneration at the boundary of a single NURBS surface. Compared with the related research in recent years, the optimized simulation algorithm of heart valves in the present invention is about 49% faster. BRIEF DESCRIPTION OF THE DRAWINGS

[0022] In order to more clearly illustrate the technical solutions in the embodiments of the present invention or the prior art, the following will briefly introduce the drawings required for the description of the embodiments or the prior art. Obviously, the following drawings are only some embodiments of the present invention. For those of ordinary skill in the art, other drawings can be obtained based on these drawings without creative efforts.

[0023] Figure 1This is a comparison of the differences between traditional isogeometric analysis and generalized isogeometric analysis from the perspectives of geometric representation, solution process, and the spline basis function space used in the fluid-structure interaction fast simulation method of heart valves based on isogeometric analysis of the present invention. Figure 1 ;

[0024] Figure 2 This is a comparison of the differences between traditional isogeometric analysis and generalized isogeometric analysis from the perspectives of geometric representation, solution process, and the spline basis function space used in the fluid-structure interaction fast simulation method of heart valves based on isogeometric analysis of the present invention. Figure 2 ;

[0025] Figure 3 This is the flowchart of the Bézier extraction algorithm for the fluid-structure interaction fast simulation method of heart valves based on isogeometric analysis of the present invention;

[0026] Figure 4 This is the schematic diagram of the piecewise representation of the geometric "copper coin" structure for the fluid-structure interaction fast simulation method of heart valves based on isogeometric analysis of the present invention;

[0027] Figure 5 This is the visualization image of the change in the velocity field of the blood flow inside the heart during the cardiac cycle for the fluid-structure interaction fast simulation method of heart valves based on isogeometric analysis of the present invention;

[0028] Figure 6 This is the visualization image of the change in the deformation of the valve leaflets during the cardiac cycle for the fluid-structure interaction fast simulation method of heart valves based on isogeometric analysis of the present invention;

[0029] Figure 7 This is the visualization image of the simulation results of heart valves represented by volume rendering for the fluid-structure interaction fast simulation method of heart valves based on isogeometric analysis of the present invention;

[0030] Figure 8 This is the flowchart of the fluid-structure interaction fast simulation method of heart valves based on isogeometric analysis of the present invention. Detailed implementation manners

[0031] Next, the technical solutions in the embodiments of the present invention will be clearly and completely described in conjunction with the accompanying drawings in the embodiments of the present invention. Obviously, the described embodiments are only a part of the embodiments of the present invention, rather than all the embodiments. All other embodiments obtained by those of ordinary skill in the art based on the embodiments of the present invention without creative efforts shall fall within the protection scope of the present invention.

[0032] In order to make the objectives, technical solutions, and advantages of the present invention clearer, the following provides a detailed description of the fluid-structure interaction fast simulation method of heart valves based on isogeometric analysis provided by the present invention patent in conjunction with the accompanying drawings, including the following steps:

[0033] Step 1: Optimize the fluid sub-problem, simplify the background grid of the fluid sub-problem, then generate a Bézier tetrahedron background grid based on the simplified grid, and finally solve the fluid sub-problem using the Geometrically Invariant Field Approximation (GIFT) concept;

[0034] Step 2: Optimize the solid sub-problem, and triangulate the NURBS surfaces used in the solid sub-problem into piecewise NURBS surfaces using a certain strategy;

[0035] Step 3: Use the augmented dynamic Lagrange multiplier method to couple the fluid-structure sub-problems. This method is a two-way coupling method and is essentially an improvement of the Lagrange multiplier method;

[0036] Step 4: Solve the fluid-structure coupling problem and analyze the results. By comparing with the results of traditional simulation methods, verify the correctness and effectiveness of this invention.

[0037] In this embodiment, the lumen length we used is 2.5 cm and the radius is 1.1 cm; the valve leaflets are represented by quadratic NURBS spline surfaces.

[0038] Further setting of this invention, the said Step 1 includes

[0039] 1-1 Simplify the background grid of the fluid sub-problem, and the simplified grid should maintain necessary geometric features.

[0040] 1-2 Generate a Bézier tetrahedron background grid based on the simplified grid. To be consistent with the finite element solver, this invention uses the Bézier extraction theory to express the Bernstein polynomial as a linear combination of Lagrange basis functions, so that other basis functions can be used for solving without modifying the shape functions of the existing finite element program. We first use the Lagrange basis functions to assemble the variational form of the NS equations:

[0041] B(W,∑u i L i )=(W,F)

[0042] where L i is the Lagrange basis function, and u i is the degree of freedom of the Lagrange basis function at the nodes. After the variational form is assembled, a linear system

[0043] KU L =F

[0044] This linear system has the same mathematical meaning as the linear systems in traditional finite element and isogeometric analysis, where U LDenote the degrees of freedom when using Lagrange basis functions. To solve for the degrees of freedom when using Bernstein basis functions, it is necessary to transform the matrices K and vectors F in the linear system. According to the relevant theory of Bézier extraction, the transformation matrix M for K and F can be derived. Through this matrix, K and F in the original linear system can be transformed into the matrices K B and vectors F B ,

[0045] K B = MK

[0046] F B = MF

[0047] Use the matrix K B and vectors F B to form a new linear system

[0048] K B U B = F B

[0049] Solve this linear system to obtain the degrees of freedom U of the Bernstein basis functions B , but since the basis used in the finite element program is still the Lagrange basis, we also need to map the degrees of freedom U B back to the Lagrange degrees of freedom through the transformation matrix M, that is, the final degrees of freedom U = M -1 U B . However, the difference here is that the degrees of freedom of the Lagrange basis functions at this time are not simply the coefficients when linearly combining the solutions at the nodes. It also includes the linear combination from the Lagrange basis functions to the Bernstein basis functions. Although this is different from the traditional Bézier extraction process, it is mathematically equivalent Figure 3 shows the entire process of Bézier extraction

[0050] 1-3 Solve the fluid sub-problem using the idea of geometrically invariant field approximation (GIFT). In order to control the situation where the numerical solution of the equation is distorted and oscillates due to the large Reynolds number and the dominant convective term, a certain stabilization method needs to be adopted to solve the NS equation to obtain the correct numerical solution. In previous solution methods, the streamline upwind Galerkin method (SUPG) was generally used as the numerical stabilization method. However, since it is difficult to determine the stabilization coefficient of this method, in the present invention, we use the VMS method developed by Hughes et al. According to the flow field decomposition theory, the flow field can generally be decomposed into resolvable coarse scales, resolvable fine scales, and unresolved fine scales. Inspired by the flow field decomposition theory, the VMS method assumes that the shape function of the solution of the NS equation is composed of a resolvable scale shape function and an unresolved scale shape function, and introduces the corresponding weight function, that is

[0051]

[0052]

[0053] where is the coarse scale variable, and U′, W′ are the fine scale variables; for convenience of representation, we can represent the variational form of the NS equation as

[0054] B(W,U) = (W,F)

[0055] In the formula, U = {u,p} is the unknown quantity to be solved in the NS equation, and W = {w,q} is the corresponding weight function, and its expansion can be expressed as:

[0056]

[0057] (W,F) = (w,0) Ω +(q,0) Ω

[0058] Substitute the shape functions and weight functions of different scales into the variational form of the NS equation, and two sets of equations projected onto the coarse scale weight function and the fine scale weight function can be obtained respectively

[0059]

[0060]

[0061] After that, the fine scale unknowns u′, p′ and the coarse scale unknowns can be solved respectively

[0062] Step 2: Optimize the solid sub-problem.

[0063] A further setting of the present invention, the said step 2 includes

[0064] 2-1 Discretize the NURBS surface used in the solid sub-problem into piecewise NURBS surfaces using a certain strategy.

[0065] To overcome the defect of NURBS surface degeneration at the boundary, the present invention represents the NURBS surface of each valve leaflet in a piecewise manner.

[0066] To avoid generating singular points after discretization, a structure similar to a "copper coin" is used in the discretization strategy to divide the surface. Figure 4 Shows that a semi-circular geometric model is divided into 4 sub-patches, and these sub-patches are combined to form the same semi-circular geometric representation as before discretization. Similarly, we can also represent the geometric model of the valve leaflet through the "copper coin" structure template as Figure 4 shown. Solving the strain of the valve leaflet using piecewise representation can effectively accelerate the simulation speed and achieve higher accuracy at the boundary, that is, obtain a more accurate solution at the boundary.

[0067] The geometric model of the valve leaflet after piecewise representation has fewer degrees of freedom because there is no degeneration at the boundary, that is, the same geometric valve leaflet model as before can be represented without using a large number of control points at the boundary. However, solving the deformation problem (solid sub-problem) of the piecewise valve leaflet geometry requires some modifications to the existing program and imposing constraints.

[0068] Step 3 Couples the fluid-structure sub-problems using the augmented dynamic Lagrange multiplier method.

[0069] The following will use the abbreviated forms of the solid sub-problem and the fluid sub-problem. Specifically, the time-discretized form of the fluid sub-problem is: find the velocity u n+1 ∈V and the pressure p n+1 ∈Q satisfying

[0070]

[0071] where the subscript u n indicates that this variational form depends on the velocity at the previous time step, and V and Q represent the function spaces where the velocity and pressure are located. Similarly, for the solid sub-problem, write its time-discretized form: find the displacement y n+1 ∈Y such that for all z n+1 ∈Y satisfying

[0072]

[0073] The subscript y n , It is indicated that this variational form depends on the displacement and the derivative of the displacement (velocity) at the previous time step, and Y is the function space where the displacement function is located.

[0074] To couple the two sub-problems, the DAL method imposes approximate constraints

[0075]

[0076] where Γ t is the mid-surface of the valve leaflet at time t, and φ t maps Γ0 to Γ t , n + α comes from the time discretization of the generalized method and represents the intermediate time between the time instants n and n + 1 when the constraint is applied, and the deformation velocity of the valve leaflet at the intermediate time depends on y n+1 , y n , L is the function space on Γ0. By imposing constraints on the fluid sub-problem and the solid sub-problem, we can write the DAL formula: find u n+1 , p n+1 , y n+1 and λ n+1 ∈ L such that for all v, q, z and δλ satisfy

[0077]

[0078] where β ≥ 0 is the augmented penalty coefficient, and the coupling constraint is imposed with the help of the Lagrange multiplier λ n+1 . Specifically, the DAL algorithm is characterized by updating the penalty term first at each time step, updating the Lagrange multiplier based on the penalty term, and then ignoring the Lagrange multiplier as an unknown from the DAL formula, making the problem become:

[0079] Find u n+1 , p n+1 , y n+1 such that for all v, q, z satisfy

[0080]

[0081] Then solve for λ n+1 such that for all δλ satisfy

[0082]

[0083] For r > 0, the update of λ n+1 is implicit, but if it is the L piecewise constant space and only one point is taken for integration at each point, then there is an explicit update method for λ n+1 :

[0084]

[0085] Step 4: Solve the fluid-structure interaction problem and analyze the results;

[0086] The result validity comparison of computer simulation of heart valves using the method of the present invention is based on relevant research in recent years, and the deviation calculation formula is:

[0087] ||m||2 = u benchmark -u prop

[0088] where u benchmark is the simulation result of recent research achievements, and u prop is the simulation result of the present invention.

[0089] In this embodiment, a heart lumen geometric model with a heart represented by Bézier tetrahedrons and a segmented valve leaflet model are used to solve the fluid-structure interaction problem in heart valve simulation. Figure 5 Shows several screenshots of the blood velocity changes in the valve lumen during the cardiac cycle. It can be seen from the figure that the closer to the middle part of the lumen (i.e., near the valve leaflets), the greater the velocity change, and vice versa, which is exactly caused by the regular deformation of the valve leaflets during the cardiac cycle, and also reflects the interaction between the valve leaflets and the blood flow from the side. Figure 6 Shows several screenshots of the deformation state of the valve leaflets during the cardiac cycle. It can be seen that the valve leaflets are initially in the closed state and gradually open over time. Combining the blood flow velocity field in the heart lumen and the deformation process of the valve leaflets, the overall changes inside the heart during a heartbeat cycle can be obtained. Figure 7 Are several screenshots of the dynamic changes of the heart valve during a cardiac cycle. We can simultaneously observe the fluid-structure interaction between the blood flow and the valve leaflets in the heart valve.

[0090] According to the experimental data in this embodiment, it can be calculated that the speed of solving the fluid-structure interaction problem of the heart valve using the method of the present invention has increased by approximately 48%.

[0091] The above has described the embodiments of the present invention in detail with reference to the accompanying drawings, but the present invention is not limited to the described embodiments. For those skilled in the art, without departing from the principle and spirit of the present invention, various changes, modifications, substitutions, and variations to these embodiments still fall within the protection scope of the present invention.

Claims

1. A fast fluid-structure interaction simulation method for heart valves based on isogeometric analysis, characterized in that: It includes the following steps: Step 1: Optimize the fluid sub-problem: Simplify the background mesh of the fluid sub-problem, generate a Bézier tetrahedron background mesh based on the simplified mesh, and solve the fluid sub-problem using the geometric invariant field approximation idea; The specific steps of Step 1 include: Simplify the background mesh of the fluid sub-problem. The simplified mesh maintains the necessary geometric features. Generate a Bézier tetrahedron background mesh based on the simplified mesh. Utilize the Bézier extraction theory to express the Bernstein polynomial as a linear combination of Lagrange basis functions, and achieve the solution using other basis functions without modifying the shape functions of the existing finite element program. Assemble the variational form of the NS equation using the Lagrange basis functions: B(W,∑u i L i )=(W,F) where L i is the Lagrange basis function, and u i is the degree of freedom of the Lagrange basis function at the node. After the variational form is assembled, a linear system is obtained: KU L = F where U L represents the degrees of freedom when using Lagrange basis functions. Transform the matrices K and vectors F in the linear system. According to the Bézier extraction theory, derive the transformation matrix M for K and F, and convert the original K and F in the linear system into the matrix K B and the vector F B : K B = MK F B = MF Use matrix K B and vector F B to form a new linear system K B U B = F B Solving this linear system, the degrees of freedom U of the Bernstein basis functions can be obtained. B , and then the degrees of freedom U B are mapped back to the Lagrangian degrees of freedom through the transformation matrix M, that is, the final degrees of freedom U = M -1 U B ; Solve the fluid sub-problem using the geometric invariant field approximation idea. The shape function of the solution of the NS equation is composed of the solvable scale and the unsolvable scale shape functions, and the corresponding weight functions are introduced, that is where \(U = \{u, p\}\) is the unknown quantity to be solved in the NS equation, \(W = \{w, q\}\) is the corresponding weight function, and its expansion can be expressed as: (W,F) = (w,0) Ω + (q,0) Ω Substitute the shape functions and weight functions of different scales into the variational form of the NS equation to obtain two sets of equations projected onto the coarse-scale weight function and the fine-scale weight function respectively Subsequently, the fine-scale unknowns u′, p′ and the coarse-scale unknowns can be solved separately Step 2: Optimize the solid sub-problem: Use the strategy of dividing the NURBS surface used in the solid sub-problem into piecewise NURBS surfaces; Step 3: Use the augmented dynamic Lagrange multiplier method to couple the fluid-solid sub-problem; Step 4: Solve the fluid-solid coupling problem and analyze the results.

2. A rapid simulation method for fluid-structure interaction of heart valves based on isogeometric analysis according to claim 1, characterized in that: In Step 3, the augmented dynamic Lagrange multiplier method is used to couple the fluid-solid sub-problem, and its specific form is:

3. A rapid fluid-structure interaction simulation method for heart valves based on isogeometric analysis according to claim 2, characterized in that: The specific steps of Step 3 include the following: Use the augmented dynamic Lagrange multiplier method to couple the fluid and solid sub-problems The fluid sub-problem in time-discrete form is: find the velocity u n+1 ∈ V and the pressure p n+1 ∈ Q such that where the subscript u n indicates that the variational form depends on the velocity at the previous time step, and V and Q represent the function spaces for the velocity and pressure, respectively; The solid subproblem in time-discrete form: find the displacement y n+1 ∈ Y such that for all z n+1 ∈ Y satisfies The subscript y in the formula n , indicates that this variational form depends on the displacement and the derivative of the displacement (velocity) at the previous time step, and Y is the function space where the displacement function is located; The two sub-problems are coupled together, and the DAL method imposes an approximate constraint where Γ t is the mid-surface of the valve leaflet at time t, φ t maps Γ0 to Γ t , n + α is the time discretization from the generalized method, representing the intermediate time between the imposed constraint times n and n + 1, and the deformation velocity of the valve leaflet at the intermediate time depends on y n+1 , y n , L is the function space on Γ0, Apply constraints to the fluid sub-problem and the solid sub-problem, and write out the DAL formula: find u n+1 , p n+1 , y n+1 and λ n+1 ∈ L such that for all v, q, z and δλ satisfy where β≥0 is the augmented penalty coefficient, and the coupling constraint is imposed by means of the Lagrange multiplier λ n+1 ​ The Lagrange multiplier is updated based on the penalty term, and then the Lagrange multiplier is ignored as an unknown quantity from the DAL formula, making the problem become: Find u n+1 , p n+1 , y n+1 such that for all v, q, z it holds that Solve for λ later n+1 such that all δλ satisfy For r > 0, the update of λ n+1 is implicit. With L piecewise constant space and integrating at only one point at each point, then λ n+1 has an explicit update method:

4. A rapid fluid-structure interaction simulation method for heart valves based on isogeometric analysis according to claim 1, characterized in that: Compare the result validity of the computer simulation of the heart valve, and the deviation calculation formula is: ||m||2 = u benchmark -u prop Among them, u benchmark is the reference result, and u prop is the simulation result.

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