CN116187147A - Near-field dynamics-based super-elastomer semi-implicit iteration simulation method - Google Patents

Near-field dynamics-based super-elastomer semi-implicit iteration simulation method Download PDF

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CN116187147A
CN116187147A CN202211717942.2A CN202211717942A CN116187147A CN 116187147 A CN116187147 A CN 116187147A CN 202211717942 A CN202211717942 A CN 202211717942A CN 116187147 A CN116187147 A CN 116187147A
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何小伟
卢子璇
刘学慧
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Abstract

The invention discloses a super-elastic body semi-implicit iteration simulation method based on near-field dynamics, which is a simulation method with high precision, high robustness and high performance, uses a near-field dynamics theory to perform energy modeling of super-elastic materials, performs dimension expansion of particle neighborhood on a residual dimension object, uses a semi-implicit iteration method to perform elastic solution, can ensure convergence, and stably processes the overshoot problem in the iteration process through self-adaptive step length. The invention can simulate the dynamic process and interaction behavior of various super-elastic materials, super-elastic cloth, thin shell and other residual dimension materials in real time with high precision and high robustness.

Description

Near-field dynamics-based super-elastomer semi-implicit iteration simulation method
Technical Field
The invention belongs to the field of computer graphics, and particularly relates to a super-elastomer semi-implicit iteration simulation method based on near-field dynamics.
Background
Superelastic objects have very unique material properties, with stress-strain relationships and strain-geometric relationships often having non-linear characteristics. Therefore, efficient and stable simulation of three-dimensional superelastic bodies, cloth and thin shells and other residual dimension superelastic bodies is an important problem in the field of graphic physics simulation. Although the finite element method based on the elastic stress method can provide accurate super-elastomer simulation solution, the method is dependent on a specific linearization method, assembly of a global stiffness matrix and sparse linear system solution, so that the requirements of efficient and real-time graphic physical simulation cannot be met. The dynamic generalized constraint is re-described by using the particle position by the position-based simulation method, and the global constraint linear system is solved by using the iterative method, so that the method has high efficiency and instantaneity, but because the constraint description of the dynamic equation is focused, the control equation does not have real physical significance, and therefore, the accurate simulation of the specific material parameters cannot be satisfied.
The reasons for the failure of the traditional super-elastomer simulation in accuracy and efficiency are mainly two aspects: (1) The implicit time integration method based on the traditional finite element relates to global rigidity matrix assembly and large-scale sparse linear system solving, and is difficult to directly parallelize due to the existence of global quantity, and the linearization method based on the Newton method relates to the solving of the Hemson matrix, so that the calculation is extremely time-consuming, and the method does not have high efficiency and real-time performance; (2) The position-based method uses particle position constraint to describe a control equation and jacobian iteration or Gaussian-Saidel iteration solution, stress calculation and linearization are not needed, and simulation can be efficiently performed, but because generalized constraint is directly assembled into a linear system to solve, inconsistency of rigidity among the constraint can cause a sparse linear system to have pathogenicity, so that for certain material parameters, convergence or larger error cannot be guaranteed.
Disclosure of Invention
The invention solves the technical problems: aiming at the problem that the traditional simulation method can not efficiently and accurately simulate three-dimensional and two-dimensional Yu Weishu super-elastic objects, the semi-implicit iteration simulation method of the super-elastic body based on near-field dynamics, which has high precision, high robustness and high performance, is provided. The method belongs to the particle method, so that the problem that the finite element method operation based on grid dispersion is not efficient can be avoided; the problem of overshoot is solved using a semi-implicit solver that guarantees convergence while using an adaptive step size. The technology provides a unified solving scheme based on superelastic energy modeling and iterative solving for the three-dimensional object and the two-dimensional rest number object, and finally can realize high-efficiency, accurate and high-robustness simulation of the superelastic body.
The invention relates to a high-precision, high-robustness and high-performance simulation method, which uses a near-field dynamics theory to perform energy modeling of a super-elastic material, introduces virtual particle bonds to a residual dimension object by using a vertex normal direction to perform dimension expansion of a particle neighborhood, solves the degradation problem of a particle shape matrix and a deformation gradient matrix in the residual dimension object simulation, and provides a unified solving scheme of a two-dimensional object and a three-dimensional object. The invention uses a semi-implicit iteration method to carry out elastic solution, decomposes the Coriolis-Cristolous stress tensor into a positive definite term and a negative definite term to construct an iteration which ensures convergence, and introduces a self-adaptive step length to solve the overshoot problem in the global iteration convergence process. The invention can efficiently simulate the dynamic process and interaction behavior of various super-elastic materials, super-elastic cloth, thin shell-shaped objects and other residual dimension materials in real time, and has good parallelizable characteristic because the assembly of a global linear system and the operation of an inner product are not involved.
The specific technical scheme of the invention is as follows:
a super-elastomer semi-implicit iteration simulation method based on near-field dynamics, as shown in figure 2, comprises the following steps:
step 1: reading grid parameters of a three-dimensional geometrical model of a simulation object, initializing the vertex position of the grid model as the particle position of the near-field dynamics super-elastomer particles, initializing the particle speed to be zero, establishing a particle neighborhood relation according to grid topology, and storing the particle position at the initial moment as a static shape;
step 2: if the current simulation time step is not the initial simulation time step, updating the position and the speed of the particles by using the result of the superelastic solver and the result of the time integrator of the last simulation time step, wherein the obtained result is the result of the last simulation time step and can be used for outputting or visualizing; using a time integrator to perform time integration on an external force field (such as gravity, load and the like) applied to a simulation object appointed by a user, updating the current particle speed and position, and obtaining a time integrator result of a current simulation time step as an input of a super-elasticity solver;
step 3: calculating a shape matrix and a deformation gradient matrix of the particles according to the current particle position, the particle speed and the particle neighborhood relation and the static shape stored in the step 1;
step 4: decomposing the Coriolis-Cristolochia stress tensor into a positive term Xiang Yufei according to a specified super-elastomer material constitutive model, and calculating a positive term Xiang Yufei of the tensor according to the particle shape matrix and the deformation gradient matrix obtained in the step 3;
step 5: according to the results obtained in the step 3 and the step 4, calculating a sparse linear system coefficient matrix and a source term in a jacobian iterative form of implicit time integration;
step 6: carrying out Jacobian iteration of a set step number according to the coefficient matrix and the source item obtained in the step 5, solving a related sparse linear system, and calculating to obtain the particle position after the iteration;
step 7: calculating the position gradient of the particles according to the difference between the particle positions obtained after one iteration and the particle positions before the iteration in the step 6, calculating the self-adaptive iteration step length capable of avoiding the overshoot problem by using the one-dimensional elasticity of the super-elastic material, and updating the particle positions by using the self-adaptive step length and the position gradient;
step 8: repeating the steps 3 to 7 and increasing the iteration step number until the position gradient residual error is smaller than the set error or the algorithm reaches the appointed iteration step number, and obtaining a result which is a superelasticity solver result;
step 9: and (3) increasing the simulation time step, and repeating the operations between the steps 2-8 until the termination time or the set simulation termination condition is reached, so as to complete the whole simulation process.
In the step 3, if the simulated object is a two-dimensional residual dimension object, dimension expansion is performed on the super-elastomer particles, and the operation is as follows:
(1) at initialization, the angle weighted average normal of the vertex i where the particle is located is pre-calculated and stored
Figure BDA0004026780640000031
Where J represents the triangle sequence number adjacent to vertex i, ζ represents the thickness of the boundary layer to the neutral layer outside the two-dimensional residual dimension object, n J Represents the external normal of triangle J, θ J Representing the interior angle of triangle J at vertex i;
(2) each simulation time step, calculating the angle weighted average normal of the vertexes i of the deformed particles
Figure BDA0004026780640000032
Wherein superscript denotes the corresponding amount in the deformed configuration;
(3) the shape matrix of particle i is expressed as
Figure BDA0004026780640000033
Deformation gradient matrix is expressed as +.>
Figure BDA0004026780640000034
The position of the particles in the undeformed initial configuration is denoted by x i Indicating that particle i is in a deformed initial configuration; use->
Figure BDA0004026780640000035
Representing the deformation configuration position of the particle i in the iterative step k; j represents an adjacent particle of the particles; omega ij Represents the scalar weight between particles i, j; />
Figure BDA00040267806400000320
Representing the kronecker product.
In the step 4, the decomposition format of the Pi Liao-kristolon stress tensor is as follows:
Figure BDA0004026780640000036
Figure BDA0004026780640000037
wherein->
Figure BDA0004026780640000038
Deformation gradient matrix is decomposed by standard SVD>
Figure BDA0004026780640000039
The obtained diagonal matrix, U i And V i Decomposing a corresponding left rotation matrix for the SVD; />
Figure BDA00040267806400000310
Diagonal matrix for using deformation gradient +.>
Figure BDA00040267806400000311
A diagonal portion of the calculated coriolis-kristolo stress tensor; />
Figure BDA00040267806400000312
A positive part of the stress tensor diagonal matrix; />
Figure BDA00040267806400000313
Is the negative part of the stress tensor diagonal matrix.
In the step 5, the operation is as follows:
(1) the jacobian iterative form of implicit time integration is expressed as:
Figure BDA00040267806400000314
(2) the coefficient matrix of the sparse linear system is expressed as:
Figure BDA00040267806400000315
Figure BDA00040267806400000316
(3) the source term of the sparse linear system is expressed as:
Figure BDA00040267806400000317
Figure BDA00040267806400000318
in the above formulae, y i 、y j In the form of iterationThe position of the particles i, j to be solved;
Figure BDA00040267806400000319
representing the deformation configuration position of the particle i in the iterative step k; />
Figure BDA0004026780640000041
Representing the deformation configuration position of the particle i at the simulation time step t; />
Figure BDA0004026780640000042
Representing the velocity of particle i at simulation time step t; m is m i Is the mass of particle i; h is the time step; scalar V i ,V j The volumes of particles i, j, respectively;
Figure BDA0004026780640000043
wherein r is i The spherical neighborhood radius of the particle i; i represents an identity matrix.
In the step 6, the jacobian iterative process operates as follows:
(1) the sparse linear system solved in the jacobian iterative process is as follows: for any of the particles i,
Figure BDA0004026780640000044
Figure BDA0004026780640000045
(2) solving using iterative methods, wherein initialization
Figure BDA0004026780640000046
The iterative process is
Figure BDA0004026780640000047
Until the residual converges or m reaches a specified maximum number of iteration steps.
In the step 7, the operation of obtaining the self-adaptive step length and updating the particle position is as follows:
(1) calculating an adaptive step length for avoiding the overshoot problem:
Figure BDA0004026780640000048
wherein E is an energy density function;
(2) updating particle positions using an adaptive step size:
Figure BDA0004026780640000049
the invention also provides a server comprising a memory and a processor, the memory storing a computer program configured to be executed by the processor, the computer program comprising instructions for performing the steps of the above method.
The invention also provides a computer readable storage medium having stored thereon a computer program, characterized in that the computer program when executed by a processor realizes the steps of the above method.
Compared with the existing method, the method has the following advantages:
(1) According to the invention, the near-field dynamics theory is used for modeling the super-elastic material, and virtual keys are introduced into the two-dimensional residual dimension object to carry out dimension expansion of the particle neighborhood, so that the problem of matrix degradation in the residual dimension object simulation is solved, and a unified solution scheme of the three-dimensional and two-dimensional Yu Weishu super-elastic object is provided;
(2) The invention provides a semi-implicit iterative solver, which can ensure global convergence by decomposing a stress matrix and assembling corresponding iterative solvers. The method is not dependent on other linearization methods and the calculation of global quantity, so that the method has the characteristics of high efficiency and real time;
(3) The whole proposal provided by the invention has very high stability, solves the overshoot problem in the iterative convergence process through the self-adaptive step length, can simulate the super-elastic material with large-scale rigidity, and can be widely applied to various super-elastic material constitutive models and user-defined materials, thereby simulating the required material properties;
(4) Each step in the invention has high parallel characteristic, thus being suitable for modern GPU architecture.
Drawings
FIG. 1 is a flow chart of an implementation of the present invention.
FIG. 2 is a schematic diagram of a unified solver.
Detailed Description
The foregoing objects, features and advantages of the invention will be more readily apparent from the following detailed description of the invention taken in conjunction with the accompanying drawings, which are not to be construed as limiting the invention.
1. The hardware platform of the method adopts eight-core CPU with model number of Intel i9-10900K, main frequency of 3.5GHz, NVIDIAGeForce RTX A4000 video card and video memory of 12GB. The system program is written in C++, wherein CUDA language is adopted for accelerating the parallel computing part, the program is compiled and executed by Microsoft Visual Studio 2019, and open source libraries such as OpenGL, eigen and the like are adopted in the development process.
2. The invention takes near field dynamics as a theoretical basis, and the specific form is as follows:
consider first the implicit time integration formula:
Figure BDA0004026780640000051
wherein the method comprises the steps of
Figure BDA0004026780640000052
Representing the position of particle i after deformation at time step t +.>
Figure BDA0004026780640000053
Represents the speed of particle i at time step t,/-, at time t>
Figure BDA0004026780640000054
Represents the indirect speed, M i Is the mass matrix of the particles, h is the time step. Reference position x of a given particle i i And the neighborhood { j }, under the near field dynamics framework, the inter-particle internal force f is determined by the particle bonds:
Figure BDA0004026780640000055
wherein the method comprises the steps ofTAs a stress state basis function, V is the volume, and the near field dynamics constitutive model is expressed as:
Figure BDA0004026780640000056
where P is the first Coriolis-Cristolous stress tensor as a function of the deformation gradient matrix F, where F is expressed as:
Figure BDA0004026780640000057
/>
wherein the method comprises the steps of
Figure BDA0004026780640000058
Thus, the implicit integration formula may be transformed from the above formula:
Figure BDA0004026780640000061
wherein the external force
Figure BDA00040267806400000612
Regardless of the superelastic solution, it can be pre-computed at the time integrator, so after transformation, the superelastic solver finally uses the following formula as the implicit time integration formula to be solved:
Figure BDA0004026780640000062
3. FIG. 1 is a flow chart of a semi-implicit iteration simulation solver of a super-elastomer based on near-field dynamics, wherein in an initialization stage before simulation, a three-dimensional geometric model is firstly read, near-field dynamics super-elastomer particles are correspondingly built one by one according to grid vertexes of the model, and bits of an initial reference form of the particles are storedPut x i Establishing and storing an adjacency relation { j } for each particle i according to a topological constraint, preprocessing the topological relation, including the relation between the voxel (triangular grid or tetrahedron grid) where the particle corresponds to the vertex and the relation between the vertices contained in the voxel, and setting an initial time step t=0.
4. After the simulation starts, if the time step t=0 is not the initial time step, firstly updating the position of the particle solved by the superelastic solver of the previous step into the current position, and using the position difference between the superelastic solver of the previous step and the time integrator divided by the time step as the speed increment of the previous step to update the current speed of the particle, wherein the variables meet the implicit time integral equation and can be further used for output or visualization; performing time integration: before each frame of super-elastic solution starts, using gravity as an external force to update the speed of the particle at the next moment, and using the speed to update the position of the particle at the next moment, wherein the pre-calculated speed and position are used as inputs of a super-elastic iterative solver, and the above process can be expressed as follows:
(1) if t is not equal to 0, updating the current position to be the solving result of the super-elastic solver:
Figure BDA0004026780640000063
(2) updating the intermediate speed and the intermediate position pre-calculated by the time integrator:
Figure BDA0004026780640000064
increment t.
5. Performing superelastic solution, setting iteration step k=0 in the initial stage, and setting initial position as the result of time step t moment for any particle i, namely initializing
Figure BDA0004026780640000065
Each iteration step k thereafter, a corresponding strain measure is calculated:
(1) for the three-dimensional super-elastomer, for each particle i, a shape matrix and a deformation gradient matrix are calculated:
Figure BDA0004026780640000066
Figure BDA0004026780640000067
(2) for two-dimensional residual dimension object, for each particle i, first calculating reference vertex normal of particle corresponding vertex
Figure BDA0004026780640000068
Normal to the current vertex: />
Figure BDA0004026780640000069
The triangle where the corresponding vertex of the particle is located is preprocessed into a topological relation, so that each particle i can be independently inquired, and then the non-degenerate form of the shape matrix and the deformation gradient matrix after the particle neighborhood dimension expansion is calculated: />
Figure BDA00040267806400000610
Figure BDA00040267806400000611
Calculating deformation gradient F i After that, SVD decomposition is performed thereon: />
Figure BDA0004026780640000071
6. Diagonal form using deformation gradients
Figure BDA0004026780640000072
Calculating a decomposed version of the Coriolis-Cristolous stress tensor:
(1) for three-dimensional superelastomers, neo-Hookean superelastic models are used
Figure BDA0004026780640000073
Figure BDA0004026780640000074
The diagonal form of the Pi Liao stress tensor is expressed as: />
Figure BDA0004026780640000075
Wherein->
Figure BDA0004026780640000076
I in the above 1 ＝tr(F T F)，I 2 ＝F T F:F T F,1 3 ＝det(F T F)；
(2) For a two-dimensional residual dimension object, the tensile strain energy and the bending strain energy are used for modeling of a constitutive model respectively, and the tensile strain energy is expressed as:
Figure BDA0004026780640000077
wherein->
Figure BDA0004026780640000078
And selecting n=3, and expressing the diagonal form of the coriolis stress tensor as: />
Figure BDA0004026780640000079
Wherein the method comprises the steps of
Figure BDA00040267806400000710
Figure BDA00040267806400000711
The bending stress state basis function is expressed as
Figure BDA00040267806400000712
Figure BDA00040267806400000713
The coefficient matrix will directly contribute to the implicit time integration formula.
7. Computing a sparse linear system coefficient matrix and a source term of jacobian iteration:
(1) for a three-dimensional superelastic object, for any particle i, the coefficient matrix is calculated as follows:
Figure BDA00040267806400000714
the source term is expressed as:
Figure BDA00040267806400000715
(2) for a two-dimensional residual dimension object, for any particle i, the coefficient matrix is calculated as follows:
Figure BDA00040267806400000716
the source term is expressed as:
Figure BDA00040267806400000717
Figure BDA00040267806400000718
8. and carrying out jacobian iteration of the specified maximum iteration step number on each particle i, wherein the initialization condition is as follows:
Figure BDA00040267806400000719
the iterative formula is:
Figure BDA00040267806400000720
9. for each particle i, a step size is calculated that avoids the overshoot problem:
Figure BDA00040267806400000721
updating the particle position using the step size is: />
Figure BDA00040267806400000722
Increment k.
10. And (3) returning the result to the time integrator to enter the loop again after the superelastic iteration solver converges or the iteration step k reaches the maximum iteration step number, resetting the temporary variable, starting stepping, and entering the calculation of the next frame until the simulation is finished.
In the invention, the calculation process and the iteration process of the correlation quantity of the particle i can be carried out on the GPU in parallel, the super-elastic constitutive model can be replaced by a corresponding constitutive according to actual requirements, and the calculation is carried out according to the decomposition method of the invention.
The above examples are only for illustrating the technical solution of the present invention and not for limiting it, and those skilled in the art may modify or substitute the technical solution of the present invention without departing from the spirit and scope of the present invention, and the protection scope of the present invention shall be defined by the claims.

Claims (8)

1. A super-elastomer semi-implicit iteration simulation method based on near field dynamics comprises the following steps:
1) Reading in a three-dimensional geometric model of an object to be simulated, wherein the three-dimensional geometric model is a grid model, taking the vertex position of the three-dimensional geometric model as the particle position of the near-field dynamics super-elastomer particles, initializing the particle speed to be zero, establishing a particle neighborhood relation according to the grid topology of the three-dimensional geometric model, and storing the particle position at the initial moment as a static shape;
2) If the current simulation time step is not the initial simulation time step, updating the position and the speed of the particles by using the simulation result of the last simulation time step; then using a time integrator to perform time integration on the external force field applied to the simulation object appointed by the user, and updating the particle speed and the position;
3) Calculating a shape matrix and a deformation gradient matrix of the particles according to the current particle position, the particle speed and the particle neighborhood relation and the static shape stored in the step 1);
4) Decomposing the Coriolis-Cristolochia stress tensor into a positive definite Xiang Yufei positive term according to a specified super-elastomer material constitutive model, and calculating the positive definite Xiang Yufei positive term according to the shape matrix and the deformation gradient matrix obtained in the step 3);
5) According to the results obtained in the steps 3) and 4), calculating a sparse linear system coefficient matrix and a source term in a jacobian iterative form of implicit time integration;
6) Carrying out Jacobian iteration solution on the sparse linear system according to the coefficient matrix and the source term obtained in the step 5), and calculating to obtain the particle position after the iteration;
7) Calculating the position gradient of the particles according to the difference between the particle positions obtained in the step 6) and the particle positions before the iteration, calculating the self-adaptive iteration step length by using the one-dimensional elasticity of the super-elastic material, and updating the particle positions by using the self-adaptive step length and the position gradient;
8) Iteratively executing the steps 3) to 7) until the gradient residual error of the position gradient is smaller than the set error or the iteration number reaches the designated iteration step number;
9) And (3) increasing the simulation time step, and repeating the steps 2) to 8) until the termination time or the set simulation termination condition is reached, so as to complete the simulation process.
2. The method according to claim 1, wherein if the object to be simulated is a two-dimensional residual dimension object, the super-elastomer particles are subjected to dimension expansion, specifically comprising: at initialization, the angle weighted average normal of the vertex i where the particle is located is pre-calculated and stored
Figure FDA0004026780630000011
Calculating the angle weighted average normal of the vertexes i of the deformed particles after each simulation time step after initialization>
Figure FDA0004026780630000012
The shape matrix of particle i is expressed as +.>
Figure FDA0004026780630000013
Deformation gradient matrix is expressed as +.>
Figure FDA0004026780630000014
x represents the position of the particle in the undeformed initial configuration, x i Indicating that particle i is in a deformed initial configuration; />
Figure FDA0004026780630000015
Representing the deformation configuration position of the particle i at the simulation time step k; j represents an adjacent particle to particle i; omega ij Representing scalar weights between particles i, j；/>
Figure FDA0004026780630000016
Representing the kronecker product.
3. The method of claim 2, wherein the decomposition format of the coriolis-kristolochia stress tensor is:
Figure FDA0004026780630000021
wherein (1)>
Figure FDA0004026780630000022
To decompose deformation gradient matrix by SVD
Figure FDA0004026780630000023
The obtained diagonal matrix, U i And V i Decomposing a corresponding left rotation matrix and a corresponding right rotation matrix for SVD; />
Figure FDA0004026780630000024
Diagonal matrix for using deformation gradient +.>
Figure FDA0004026780630000025
A diagonal portion of the calculated coriolis-kristolo stress tensor; />
Figure FDA0004026780630000026
A positive part of the stress tensor diagonal matrix; />
Figure FDA0004026780630000027
Is the negative part of the stress tensor diagonal matrix.
4. A method according to claim 1, 2 or 3, wherein the specific implementation method of step 5 is as follows: the jacobian iterative form of implicit time integration is expressed as:
Figure FDA0004026780630000028
coefficient matrix of sparse linear system +.>
Figure FDA0004026780630000029
Source of sparse linear System->
Figure FDA00040267806300000210
y i 、y j The positions of particles i and j to be solved in an iterative form are obtained; />
Figure FDA00040267806300000211
Representing the deformation configuration position of the particle i in the iterative step k; />
Figure FDA00040267806300000212
Representing the deformation configuration position of the particle i at the simulation time step t; />
Figure FDA00040267806300000213
Representing the velocity of particle i at simulation time step t; m is m i Is the mass of particle i; h is the time step; scalar V i ,V j The volumes of particles i, j, respectively; />
Figure FDA00040267806300000214
Wherein r is i The spherical neighborhood radius of the particle i; i represents an identity matrix.
5. A method according to claim 1, 2 or 3, characterized in that the adaptation step size of the particles i at iteration step k+1 is
Figure FDA00040267806300000215
Wherein E is an energy density function, +.>
Figure FDA00040267806300000216
Representing the deformation configuration position of the particle i in the iteration step k, y i The position of the particle i to be solved; particle i particle position updated with adaptive step size at iteration step k+1 is +.>
Figure FDA00040267806300000217
6. A method according to claim 1 or 2 or 3, wherein the simulation results comprise a superelastic solver result and a time integrator result; the time integrator results in that the time integrator is used for performing time integration on an external force field appointed by a user, and the current particle speed and the current particle position are updated; and 3) taking the time integrator result as the input of the super-elastic solver to execute the results obtained in the steps 3) to 7) to obtain the super-elastic solver result.
7. A server comprising a memory and a processor, the memory storing a computer program configured to be executed by the processor, the computer program comprising instructions for performing the steps of the method of any of claims 1 to 6.
8. A computer readable storage medium, on which a computer program is stored, characterized in that the computer program, when being executed by a processor, implements the steps of the method of any of claims 1 to 6.
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