CN113076677B - Flexible body structure high-order nonlinear finite element numerical simulation method based on quintic hermite function - Google Patents
Flexible body structure high-order nonlinear finite element numerical simulation method based on quintic hermite function Download PDFInfo
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Abstract
The invention discloses a novel method for numerical simulation of a high-order nonlinear finite element of a flexible body structure based on a quintic Hermite shape function, which is used for deducing the quintic Hermite interpolation shape function and an attribute equation based on a quintic Hermite high-order interpolation theory; determining a flexible body structure unit and parameters, a physical model and simplifying assumptions which need to be solved by the flexible body structure unit; establishing a mathematical motion partial differential equation of the flexible body structure unit in space and time and a stretching limiting condition along the axial direction, namely solving a mathematical model in a domain; thereby establishing a system equation of the flexible body structure unit and adopting a Newton iteration method to carry out iteration solution on the parameters. Has the advantages that: the simulation precision of the space deformation, the corner and the curvature of the slender flexible body structure is improved; the continuity of the curvature at the finite element node is ensured; the fast convergence is realized, and particularly, the prediction precision of the high-order vibration frequency of the flexible body is higher; when the problems of transverse large gradient load and large bending of the flexible body are solved, higher precision guarantee is provided.
Description
Technical Field
The invention relates to the technical field of flexible body (slider structures) simulation, in particular to a high-order nonlinear finite element numerical simulation method of a flexible body structure based on a quintic Hermite shape function.
Background
Up to now, the most commonly used finite element method for flexible body structural elements (such as beam elements) in the world is still a finite element method based on cubic Hermite shape functions (cubic Hermite shape functions). The method can only ensure the continuity of the space configuration (spatial configuration) of the flexible body and the first derivative (namely the corner slope angle) of the configuration in the discrete nodes of the finite element.
Since the spatial form of the flexible body unit structure is simulated and segmented into a cubic polynomial function, the first derivative is a spatial quadratic function, namely, quadratic distribution, and the second derivative is a spatial linear function, namely, linear distribution. Although this method has achieved good numerical simulation accuracy, it has some inherent drawbacks: the simulated result of the method can only ensure the continuity of a first derivative function of the space form of the space flexible structure at a finite element node, but cannot ensure the continuity of a second derivative function; and the Dirichlet (Dirichlet) boundary condition cannot be applied for the flexible body cell boundary condition with zero curvature (curvature), resulting in deviation (deviation) of the physical model and the numerical model boundary condition; when the flexible body is subjected to a large gradient load (namely, when the distribution of bending moment and shearing force inside the structure body has large mutation), the algorithm is difficult to accurately capture the positions and the sizes of extreme values of the bending moment and the shearing force (position and mass of the critical bending and shear force), and a very fine finite element unit is required to capture the extreme values.
When the flexible body structure is designed, the natural vibration frequency of the flexible body often needs to be accurately known, so that the natural frequency range of main external excitation load possibly encountered by the structure in the use process is avoided, and the negative influence caused by resonance is prevented or reduced. By adopting a cubic Hermite finite element method, the natural frequency of high-order vibration of the flexible body structure is difficult to be measured accurately, a large error exists, and the larger the unit is, the larger the error is.
Disclosure of Invention
Aiming at the problems of the traditional method, the invention provides a high-order nonlinear finite element numerical simulation method of a flexible body structure based on a quintic Hermite shape function. The quintic Hermite special-shaped function finite element method is introduced into the computer finite element numerical simulation of the flexible body structure, and the computer numerical simulation precision of the space deformation, the corner and the curvature of the slender flexible body structure is improved.
In order to achieve the purpose, the invention adopts the following specific technical scheme:
a flexible body structure high-order nonlinear finite element numerical simulation method based on a quintic hermitian function is characterized by comprising the following specific steps:
the method comprises the following steps: based on a Quintic Hermite (Quintic Hermite) high-order interpolation theory, quintic Hermite interpolation shape functions are deduced under a one-dimensional coordinate system, and the Quintic Hermite interpolation shape functions comprise 6 Quintic shape functions which are specifically as follows:
wherein the content of the first and second substances,
x a and x b Respectively as the start point coordinate and the end point coordinate of the space finite element unit node, and the coordinate interval range under the one-dimensional overall coordinate system x is x ∈ [ ] a ,x b ](ii) a The coordinate interval range of xi under the local coordinate system is xi epsilon [0,1]；
The attribute equation of the quintic Hermite interpolation shape function is as follows:
step two: determining a flexible body structure unit and parameters to be solved by the flexible body structure unit, determining a physical model of the flexible body structure unit according to size parameters of the flexible body structure unit, and establishing a mathematical motion partial differential equation of the flexible body structure unit in space and time on the basis of adopting a reasonable simplifying assumption; analyzing to obtain the deformation characteristic of the flexible body structure unit in the axial direction, and acquiring the stretching limiting condition of the flexible body structure unit along the axial direction;
the mathematical motion partial differential equation of the flexible body structure unit in space and time is concretely as follows:
wherein, M m Is a quality matrix;
r、
q is a variable which changes with the arc length s and the time t under an arc coordinate system;
wherein r is a space instantaneous position vector of the flexible body structure unit after deformation;
b is the bending rigidity of the flexible body structure unit, B = EI, E is the Young modulus of the material of the flexible body structure unit, and I is the inertia moment of the flexible body structure unit to the neutral axis;
q is the load distribution of the external transverse load along the length direction of the flexible body structure unit;
the black dot at the top of the symbol represents the derivative with respect to time, and the two dots represent the second derivative with respect to time; the' in the upper right hand corner of the symbol represents the first derivative to the spatial arc length; the "in the upper right hand corner of the symbol represents the second derivative to the spatial arc length; the "' in the upper right hand corner of the notation represents the third derivative to spatial arc length; "in the upper right hand corner of the symbol represents the fourth derivative to spatial arc length; by default, bold black indicates that the physical quantity is a vector or tensor, and non-bold indicates a scalar quantity.
Step three: substituting the quintic Hermite interpolation shape function (1) obtained in the first step into a mathematical motion partial differential equation (3) of the flexible body structure unit in space and time, integrating the whole flexible body structure unit by adopting partial integration, and establishing a nonlinear system equation on the space unit of the flexible body structure unit by combining the quintic Hermite interpolation shape function attribute equation (2) in the first step:
wherein L represents the length of the flexible body structural unit;
a i (s) represents the expression of the quintic hermitian interpolation function in the overall arc coordinate system;
a i (s)＝J i ·φ i (ξ) (i =1 to 6); establishing a conversion relation of the overall coordinate system and the form function expressions under the unit coordinate system; j is a unit of i Representing Jacobian transformation coefficients under different coordinate systems;
the relationship between subscript i and length L of the flexible body structural unit is:
step four: applying boundary conditions at two ends of the flexible body structure unit, and defining the equal sign right side of equation (5) as the generalized force F of the flexible body structure unit i Generalized force F of the flexible body structural unit i The expression is as follows:
step five: discretizing the parameters in the nonlinear system equation (5) in the third step by using a tensor algorithm:
obtaining the spatial instantaneous position distribution r (s, t) and the high-order derivative r of the flexible body structural unit by the quintic Hermite interpolation shape function approximate simulation (n) (s，t)；
By a quadratic interpolation function p m (s) obtaining effective tension distribution inside the flexible body structural unit by approximate simulation
The internal mass distribution M (s, t) of the flexible body structure unit and the internal bending rigidity distribution B(s) of the flexible body structure unit;
the specific simulation physical quantity equation is as follows:
step six: substituting the numerical simulation physical quantity equation (7) obtained in the step five into a nonlinear system equation (5) on a space unit of the flexible body structure unit, and combining the generalized force F of the flexible body structure unit in the step five i And (3) deriving a discrete equation in a flexible body structure unit by using an expression (6) and an attribute equation (2) of a quintestic interpolation shape function:
α ikm an integral coefficient matrix related to a bending stiffness term generated by the Galerkin method;
β ikm an integral coefficient matrix related to a tension term generated by the Galerkin method;
γ ikm an integral coefficient matrix related to an inertial force term generated by the Galerkin method;
q in vectors generated for external laterally distributed loads applied to the flexible body elements;
F in is the generalized force vector at the unit node;
step seven: performing coordinate system transformation on discrete equations (8) within the flexible body structural units: the coordinate system of the quintestic Hermite interpolation shape function is converted from a space overall coordinate system through a Jacobian conversion coefficient J i And (3) converting the data into a unit local coordinate system to obtain a discrete equation in the flexible body structure unit under the local coordinate system, so that the rigidity coefficient matrix in the equation (8) can be conveniently solved.
Wherein, the discrete system equation (9) in the flexible body structure unit under the local coordinate system is:
all stiffness coefficient matrices in equation (8) solve the relationship:
step eight: establishing a stretching limiting condition of the flexible body structure unit along the axial direction in a simultaneous step II, repeating the steps III to V based on a similar discrete method, discretizing the stretching limiting condition of the flexible body structure unit along the axial direction to obtain a discrete equation (11) corresponding to the stretching limiting condition of the flexible body structure unit along the axial direction;
step nine: and solving the nonlinear coupling system equation on the space unit of the flexible body structure unit by a Newton iteration method according to the discrete system equation (9) in the flexible body structure unit under the local coordinate system and the discrete equation (11) corresponding to the stretching limit condition of the equation flexible body structure unit along the axial direction.
By the design, a finite element algorithm of a quintic Hermite special function is adopted,
the curvature is brought into generalized displacement to be directly solved, so that the continuity of the curvature at a finite element node can be guaranteed, and the continuity of the bending moment of the flexible body and the continuity of bending moment induced stress in a space solution domain are guaranteed; the algorithm can solve a smaller total stiffness matrix under the condition of dividing fewer units, and compared with the traditional finite element method, the method is higher in precision and low in calculation cost; under the same condition of the same flexible body structure unit and the same thickness division, the precision of the new algorithm for solving the natural frequency of the flexible body, particularly the high-order natural frequency, is higher, so that the more accurate dynamic characteristics of the flexible body can be obtained, and the dynamic and fatigue prediction precision is improved. For solving the problem of the flexible body with large curvature, the precision of the new algorithm can be well guaranteed no matter the thickness of the divided units, and the traditional method needs extremely fine units to obtain higher precision, so that the numerical finite element of the flexible body is solved, the reliability of the numerical simulation result is better than that of the traditional finite element method no matter the thickness of the units of the calculation result of the new method, and the confidence of a decision maker on the calculation result and engineering design can be increased.
The flexible body structure unit has a high utilization ratio in engineering and shows a trend of a higher utilization ratio (because the engineering structure meets the strength condition, the use of construction materials can be effectively reduced by adopting a large number of flexible body slender structures, the structure is light and convenient to install, and the total construction cost can be greatly reduced), and the flexible body structure unit is particularly suitable for various ocean engineering projects. Such as various marine risers, pipelines, anchor chains, umbilical cables, pipe-in-pipe, multi-layer pipes, novel composite pipes, and the like; cables, high-voltage wires, optical cables, ropes of various materials; engineering beam structures of various shapes, such as bearing beams, upright beams, L-shaped/T-shaped/U-shaped/rectangular beams for buildings, long-span bridges, wings of airplanes, fan blades for wind power generation, vertical supporting uprights and the like, belong to flexible body structures.
Further, in the second step, the deformation characteristics of the flexible body structure unit in the axial direction at least include a case that the flexible body structure unit is not stretchable along the axial direction and a case that the flexible body structure unit is deformed and stretched along the axial direction;
in the case that the flexible body structure unit is not stretchable along the axial direction, the axial stretching limitation conditions are as follows:
r'·r'＝1 (4-a)
for the flexible body structure unit along the axial deformation stretching condition, the axial stretching limiting condition is as follows:
r'·r'＝(1+ε) 2 (4-b)
in equations (4-a) and (4-b), the middle black dot represents the inner product or dot product of the vectors;
epsilon represents the axial strain of the flexible body structural unit,
wherein T is the local axial tangential tension, A represents the cross-sectional area of the flexible body structure unit, and E is the Young modulus of the flexible body structure unit.
The stretch limitation conditions of the flexible body along the axial direction generally include the different situations of non-stretch along the axial direction, small deformation stretch, large deformation stretch and the like. In the present application, a large deformation tensile flexible body is not considered, and examples thereof include a rope having a large axial strain.
In a further technical solution, in step eight, for the case that the flexible body structural unit is not stretchable along the axial direction, the discrete equation corresponding to the stretching limitation condition of the axial direction is:
for the case that the flexible body structure unit has deformation and stretching along the axial direction, the discrete equation corresponding to the stretching limiting condition of the axial direction is as follows:
wherein, for example, the flexible body structure is an ocean riser, and the relational expression of the axial strain of the flexible body structure unit, the axial effective tension and the internal and external hydrostatic pressure of the riser is as follows:
A f the cross section area is corresponding to the outer diameter of the flexible body structure unit;
A c the cross section area is corresponding to the inner diameter of the flexible body structure unit;
ρ f gyA f -ρ c gyA c the pressure difference of the static water external pressure and the static water internal pressure at the position of a vertical coordinate y is corresponding to the external fluid and the internal fluid of the flexible body structure unit;
when the flexible body structure unit is not in the fluid, i.e. the environment has a small density such as air, ρ can be approximately considered f gyA f -ρ c gyA c ＝0；
The coefficient matrix in the equation can be solved by the following integral relation:
further, when the flexible body structure unit is in a static state, and a newton iteration method is used for solving the nonlinear system equation on the space unit of the flexible body structure unit, the first term, i.e., the unsteady term, of the equation (9) can be ignored, and the problem can be simplified to a greater extent. If the letter r is used to represent the number of iteration steps, the equations of the iterative algorithm for the r-th step and the r-1-th step are detailed in equations (14) and (15):
in a further technical scheme, when the flexible body structure unit is in a dynamic state, a Newton iteration method is adopted to solve a nonlinear system equation on a space unit of the flexible body structure unit, and a letter r is used for representing iteration steps. The non-constant term of equation (9) is not negligible, and the total time to be simulated in the time domain needs to be divided into a plurality of time steps s, so that the system updates the displacement, the speed and the acceleration of the flexible body after being balanced at each time step, and the updated information is used for iteration of the next time step. Is a process that moves forward with time steps and iterates continuously. For the current time step, an iterative relationship algorithm can be established to solve the axial small-deformation flexible body motion equation and solve the power problem, and the size of the time step is generally determined according to the convergence condition. Iteration is carried out at each time step (indicated by letter s), and the iterative algorithm formula of the r step and the r-1 step is detailed in equations (16) and (17):
compared with the traditional cubic Hermite special function finite element method, the method has the beneficial effects that:
the method adopts a new quintic hermitian interpolation shape function finite element algorithm (1) to bring the curvature of the flexible body into generalized displacement for direct solution, and can ensure the continuity of the curvature at a finite element node, thereby ensuring the continuity of the bending moment of the flexible body and the bending moment induced stress in a space solution domain
(2) The new algorithm can solve a smaller total stiffness matrix under the condition of dividing fewer units, and can obtain higher precision compared with the traditional cubic Hermite shape function finite element method, so that the computer computing cost can be saved.
(3) Under the condition of the same unit thickness, the precision of the new algorithm for solving the natural frequency of the flexible body, particularly the high-order natural frequency, is higher, so that the more accurate dynamic characteristic of the flexible body can be obtained, and the dynamic strength calculation and fatigue life prediction precision of the flexible body are improved.
(4) For solving the problem of the flexible body with large curvature, the precision of the new algorithm can be well guaranteed no matter the unit is divided into the thickness, and the traditional method (such as a common cubic Hermite special function finite element method) needs to require extremely fine units to obtain higher precision, so that the finite element numerical value of the flexible body is solved, the calculated result of the new method is better than that of the traditional finite element method no matter the unit is thick and the reliability is better, and the confidence of a decision maker on the structure calculation result of the flexible body and the engineering design can be increased.
Drawings
FIG. 1 is a flow chart of a method of the present invention;
FIG. 2 is an image of a quintic Hermite shape function;
FIG. 3 is a flow chart of a flexible body structure element dynamics state solution for a quintic hermitian function finite element method;
FIG. 4 is a schematic view of a simple beam model according to example 1;
FIG. 5 is a schematic diagram of the spatial configuration of the riser of example 3 before deformation;
FIG. 6 is a schematic view of the deformed riser in the embodiment 3.
Detailed Description
The following detailed description of the embodiments and the working principles of the present invention will be made with reference to the accompanying drawings.
A method for simulating a high-order nonlinear finite element numerical value of a flexible body structure based on a quintic Hermite shape function can be seen by referring to FIG. 1, and comprises the following specific steps:
the method comprises the following steps: based on a Quintic Hermite (Quintic Hermite) high-order interpolation theory, a Quintic Hermite interpolation shape function is deduced under a one-dimensional coordinate system, and a shape function image is shown in detail in figure 2; the quintic hermitian interpolation shape function comprises 6 quintic shape functions which are specifically as follows:
wherein, the first and the second end of the pipe are connected with each other,
x a and x b Respectively as the start point coordinate and the end point coordinate of the space finite element unit node, and the coordinate interval range under the one-dimensional overall coordinate system x is x E [ x ∈ [ x ] a ,x b ](ii) a The coordinate interval range of xi under the local coordinate system is xi epsilon [0,1]；
The attribute equation of the quintic Hermite interpolation shape function is as follows:
step two: determining a flexible body structure unit and parameters to be solved by the flexible body structure unit, determining a physical model of the flexible body structure unit according to size parameters of the flexible body structure unit, and establishing a mathematical motion partial differential equation of the flexible body structure unit in space and time on the basis of adopting a reasonable model hypothesis; analyzing to obtain the deformation characteristic of the flexible body structure unit in the axial direction, and acquiring the stretching limiting condition of the flexible body structure unit along the axial direction;
the mathematical motion partial differential equation of the flexible body structure unit in space and time is specifically as follows:
wherein M is m Is a quality matrix;
r、
q is a variable which changes with the arc length s and the time t under an arc coordinate system;
wherein r is a space instantaneous position vector of the flexible body structure unit after deformation;
b is the bending rigidity of the flexible body structure unit, B = EI, E is the Young modulus of the material of the flexible body structure unit, and I is the moment of inertia of the flexible body structure unit to the neutral axis;
q is the load distribution of the external transverse load along the length direction of the flexible body structure unit;
the black point at the top of the symbol represents the derivative with respect to time, and the two points represent the second derivative with respect to time; the' in the upper right hand corner of the symbol represents the first derivative to the spatial arc length; the symbol "in the upper right corner represents the second derivative to spatial arc length; the prime in the upper right hand corner of the notation represents the third derivative to spatial arc length; "in the upper right hand corner of the symbol represents the fourth derivative to spatial arc length; by default, bold and black indicates that the physical quantity is a vector or tensor, and non-bold indicates a scalar quantity.
In the second step, the deformation characteristics of the flexible body structure unit in the axial direction at least comprise the condition that the flexible body structure unit is not stretched along the axial direction and the condition that the flexible body structure unit is deformed and stretched along the axial direction;
in the case that the flexible body structure unit is not stretchable along the axial direction, the axial stretching limitation conditions are as follows:
r'·r'＝1 (4-a)
in the case of deforming and stretching the flexible body structure unit along the axial direction, the axial stretching limiting conditions are as follows:
r'·r'＝(1+ε) 2 (4-b)
in equations (4-a) and (4-b), the middle black dot represents the inner product or dot product of the vectors;
epsilon represents the axial strain of the flexible body structural unit,
wherein T is the local axial tangential tension, A represents the cross-sectional area of the flexible body structure unit, and E is the Young modulus of the flexible body structure unit.
Step three: substituting the quintic Hermite interpolation shape function (1) obtained in the first step into a mathematical motion partial differential equation (3) of the flexible body structure unit in space and time, integrating the whole flexible body structure unit by adopting partial integration, and establishing a nonlinear system equation on the space unit of the flexible body structure unit by combining the quintic Hermite interpolation shape function attribute equation (2) in the first step:
wherein L represents the length of the flexible body structural unit;
a i (s) represents the expression of the quintic hermitian interpolation function in the overall arc coordinate system;
a i (s)＝J i ·φ i (ξ) (i =1 to 6); establishing a conversion relation of the overall coordinate system and the form function expressions under the unit coordinate system;
J i representing Jacobian transformation coefficients under different coordinate systems;
the relationship between subscript i and length L of the flexible body structural unit is:
step four: applying flexible bodiesDefining the boundary conditions at two ends of the structural unit and the right side of equation (5) with equal sign as the generalized force F of the flexible body structural unit i Generalized force F of the flexible body structural unit i The expression is as follows:
step five: discretizing the parameters in the nonlinear system equation (5) in the third step by using a tensor algorithm:
obtaining the spatial instantaneous position distribution r (s, t) and the high-order derivative r of the flexible body structure unit by the quintic Hermite interpolation shape function approximate simulation (n) (s，t)；
By a quadratic interpolation function p m (s) obtaining effective tension distribution inside the flexible body structural unit by approximate simulation
Mass distribution M (s, t) inside the flexible body structure unit and bending rigidity distribution B(s) inside the flexible body structure unit;
the specific simulation physical quantity equation is as follows:
step six: substituting the numerical simulation physical quantity equation (7) obtained in the step five into a nonlinear system equation (5) on a space unit of the flexible body structure unit, and combining the generalized force F of the flexible body structure unit in the step five i And (3) deriving a discrete equation in a flexible body structure unit by using an expression (6) and an attribute equation (2) of a quintestic interpolation shape function:
α ikm an integral coefficient matrix related to a bending stiffness term generated by the Galerkin method;
β ikm an integral coefficient matrix related to a tension term generated by the Galerkin method;
γ ikm an integral coefficient matrix related to an inertial force term generated by the Galerkin method;
q in vectors generated for external laterally distributed loads applied to the flexible body elements;
F in is the generalized force vector at the unit node;
step seven: performing coordinate system transformation on discrete equations (8) within the flexible body structural units: the coordinate system of the quintestic Hermite interpolation shape function is converted from a space overall coordinate system through a Jacobian conversion coefficient J i Converting to a unit local coordinate system to obtain a discrete equation in the flexible body structure unit under the local coordinate system, and conveniently solving a rigidity coefficient matrix in an equation (8);
wherein, the discrete system equation (9) in the flexible body structure unit under the local coordinate system is:
all stiffness coefficient matrices in equation (8) are solved for the relationship:
step eight: establishing a stretching limiting condition of the flexible body structure unit along the axial direction in a simultaneous step II, repeating the steps III to V based on a similar discrete method, discretizing the stretching limiting condition of the flexible body structure unit along the axial direction to obtain a discrete equation (11) corresponding to the stretching limiting condition of the flexible body structure unit along the axial direction;
in the eighth step, for the case that the flexible body structure unit is not stretchable along the axial direction, the discrete equation corresponding to the stretching limitation condition of the axial direction is:
for the case of the flexible body structure unit deforming and stretching along the axial direction, the stretching limit condition of the axial direction corresponds to a discrete equation:
wherein, the relational expression of flexible body structure unit axial strain and effective pulling force of axial and riser inside and outside hydrostatic pressure does:
A f the cross section area is corresponding to the outer diameter of the flexible body structure unit; for flexible body structural units as risers A f The cross section area is a circular cross section area corresponding to the outer diameter of the vertical pipe structural unit;
A c the cross section area is corresponding to the inner diameter of the flexible body structure unit; for flexible body structural units as risers A c Is a circular cross-sectional area corresponding to the inner diameter of the riser.
ρ f gyA f -ρ c gyA c The pressure difference of the static water external pressure and the static water internal pressure at the position of a vertical coordinate y is corresponding to the external fluid and the internal fluid of the flexible body structure unit;
when the flexible body structure unit is not in the fluid, the correction term rho f gyA f -ρ c gyA c ＝0；
The coefficient matrix in equation (11) can be solved by the following integral relation:
step nine: and solving the nonlinear coupling system equation on the space unit of the flexible body structure unit by a Newton iteration method according to the discrete system equation (9) in the flexible body structure unit under the local coordinate system and the discrete equation (11) corresponding to the stretching limit condition of the equation flexible body structure unit along the axial direction.
When the flexible body structure unit is in a statics state, a Newton iteration method is adopted to solve the nonlinear system equation on the space unit of the flexible body structure unit, the letter r represents the number of iteration steps, and the iterative algorithm formulas of the step r and the step r-1 are detailed in equations (14) and (15):
and when the flexible body structure unit is in a dynamic state, solving a nonlinear system equation on the space unit of the flexible body structure unit by adopting a Newton iteration method, and expressing iteration step numbers by using a letter r. The non-constant term of equation (9) is not negligible, and the total time to be simulated in the time domain needs to be divided into a plurality of time steps s, so that the system updates the displacement, the speed and the acceleration of the flexible body after being balanced at each time step, and the updated information is used for iteration of the next time step. Is a process that moves forward with time steps and iterates continuously. For the current time step, an iterative relationship algorithm can be established to solve the axial small-deformation flexible body motion equation and solve the power problem, and the size of the time step is generally determined according to the convergence condition. The iteration is carried out at each time step (expressed by letter s), and the iterative algorithm formula of the r step and the r-1 step is detailed in equations (16) and (17):
specifically, referring to fig. 3, when the flexible body structural unit is in a dynamic state, the step of solving the nonlinear system equation on the space unit of the flexible body structural unit by using a newton iteration method includes:
s1: collecting all input data required by finite element numerical simulation calculation of the flexible body structure unit;
s2: initializing displacement, speed, acceleration, effective tension and all convergence control parameters of the flexible body structure unit, determining time step length and total time step, and initializing time step s =0
S3: the number of iteration steps is represented by letter r, the iteration step r =0, s = s +1 is initialized
S4: r = r +1; establishing a finite element unit stiffness array of the flexible body structure unit;
s5: all the unit stiffness arrays are assembled into a total stiffness array;
s6: applying boundary conditions and loads and numerically solving a total stiffness array;
s7: updating the displacement, speed and acceleration of the current iteration step;
s8: judging whether the displacement increment is smaller than the preset convergence precision or not; if yes, converging, and outputting a convergence result of the step s; if not, the step S9 is carried out;
s9: judging whether the current r is smaller than the maximum allowable iteration number; if yes, returning to the step S4, otherwise, diverging the program and returning to the step S2.
The algorithm provided by the invention has a wide application range, wherein the flexible structure has a high utilization ratio in engineering and shows a tendency of higher utilization ratio (because the engineering structure meets the strength condition, the use of construction materials can be effectively reduced by adopting a large number of slender structures of flexible bodies, the structure is light and convenient to install, and the total construction cost can be greatly reduced), and the method is particularly suitable for ocean engineering projects. Such as various marine risers, pipelines, anchor chains, umbilical cables, pipe-in-pipe, multi-layer pipes, novel composite pipes, and the like; cables, high-voltage wires, optical cables, signal transmission cables, ropes and the like made of various materials; various engineering beam structures, such as bearing beams, upright beams, L-shaped/T-shaped/U-shaped/rectangular beams for buildings, long-span bridges, wings of airplanes, fan blades and uprights for wind power generation and the like, all belong to slender flexible structures. By applying the novel high-order interpolation algorithm of the patent, the computer numerical simulation precision of various elongated flexible bodies can be improved, and the continuity of curvature and bending induced stress distribution in a structure in a problem solving domain is guaranteed. Meanwhile, due to the rapid convergence of the new algorithm, the calculation result is not sensitive to the division thickness of the finite element unit, so that a thicker grid can be divided under the condition of ensuring the convergence of the algorithm, and the result and the precision of the result meeting the engineering requirement can be obtained under the condition of reducing the total freedom degree of the result solution.
The invention will now be described by way of example in three embodiments:
the first embodiment is as follows: as can be seen from fig. 4, the flexible body structure unit is: a simply supported beam which is elastically supported; and solving the position and the value of the maximum bending moment of the beam of the simply supported beam with the elastic support under the action of seven distributed loads.
The parameters of the simply supported beam of the elastic support are as follows: l =8m, EI =6.9e3Nm 2 Q0=50N/m, and the rigidity of the uniformly distributed elastic supports is Ks =1.0e4N/m/m;
based on the above steps, the results of the bending moment calculation by the quintic hermite finite element new method (QH) are shown in table 1, and the results of the bending moment calculation by the conventional cubic hermite finite element method (CH) are also shown in table 1:
table 1: comparison of results of calculation of bending moment in working example 1
In the table, the Mesh Size table divides the unit thickness, and the unit length is 1m as 1; MBM represents the value of the maximum bending moment; MBM-best represents the most accurate maximum bending moment value which can be obtained; pos represents the location where the maximum bending moment occurs on the beam; rdif represents the relative error;
as can be seen from Table 1, the calculation result of the traditional cubic Hermite finite element method (CH) is sensitive to the unit division thickness, and the position of the maximum bending moment is not very accurate; the maximum bending moment value obtained by the quintic Hermite finite element new method (QH) is insensitive to the thickness of the unit division, and the position of the maximum bending moment is very accurate under the condition of the coarsest unit division. A similar conclusion can be applied to the calculation of the maximum shear force in table 2 below.
The results of calculating the maximum shear using the conventional cubic hermite finite element method (CH) and the quintic hermite finite element new method (QH) proposed by the present invention are shown in table 2:
table 2: comparison of maximum shear calculation results of example 1
In the table: the Mesh Size table divides the thickness of the unit, and the unit length is 1m as 1; MSF denotes the value of maximum shear; MSF-best represents the most accurate maximum shear value that can be obtained; rdif represents a relative error; GDoFs represents the total number of degrees of freedom required for solution.
The second embodiment: a journal paper was published in 1976 by Dareing and Huang for natural frequency calculations for a drilling riser. The specific literature names are: dareing, d.w., huang, t.,1976.Natural frequency of marine driling rise.j.pet.tech 28,813-818.
The flexible body structure unit is: a drilling riser; the flexible body structural unit size parameters are detailed in table 3;
table 3: natural frequency calculation input parameters for drilling risers
The results of the calculations to obtain the natural frequency of the drilling riser according to the above calculation steps are detailed in table 4:
table 4: comparison of results of natural frequency calculation in example 2
Considering that the natural frequency of the drilling RISER has no mathematical analytic solution, if the result of the quintic Hermite finite element new method (RISER 3D _ QH) proposed by the present invention in the case of 14 element divisions is taken as the reference standard result, the results of the old finite element method (cubic Hermite FEM) in the case of 7 and 14 element divisions and the new algorithm (quintic Hermite FEM) in the case of 7 element divisions can be compared with the reference result, and the calculated relative error is shown in table 4. As can be seen from Table 4, after the new method (RISER 3D _ QH) refines the unit, the natural frequency of the drilling RISER has little variation with respect to the error, and the percentage variation of the natural frequency of the highest order 12 th order does not exceed 1%. For the traditional finite element algorithm (CABLE 3D _ CH), when the unit is changed from thick to thin, the change of the natural frequency of 12 th order is as high as about 30 percent, which shows that the result of the traditional finite element algorithm is very sensitive to the thickness division of the unit and the convergence speed is not good; the quintic Hermite finite element new method (QH) provided by the invention has considerable precision no matter the unit thickness and is insensitive to unit thickness division.
Example three: predicting the bending moment of a shallow water Jumper after the shallow water Jumper transversely translates on a top platform;
this example refers to the international journal paper published in Connaire, et al, (2015), which is collectively referred to as: advances in the subsea roller analysis using quasi-ratios and the Newton-Raphson method; as can be seen from the literature Advances in the sub-sea prism analysis using quasi-ratios and the Newton-Raphson method, the main input parameters are shown in tables 5 and 6:
TABLE 5 input parameters for simulating deformation of flexible pipe when large transverse translation of top platform occurs
TABLE 6 hypothetical parameters for simulating deformation of a flexible pipe during large lateral translations of the top end platform
TABLE 7 comparison of bending moment calculations
Referring to fig. 5, the present example mainly simulates the large bending deformation of a flexible riser after the top end of the flexible riser (i.e., the origin of coordinates of fig. 5) moves horizontally to the right 15 meters along the x direction, and compares how the maximum bending moment inside the riser changes when different unit thicknesses are divided by using the conventional finite element algorithm (CH) and the quintessential finite element new method (QH) proposed by the present invention.
Table 7 shows the maximum bending moment comparison of the two algorithms, the conventional cubic hermite finite element method (CH) and the quinhermite finite element method (QH), for different element thicknesses when the riser is moved to the position of FIG. 6.
As can be seen from the results shown in table 7, if the calculation result of the quintic hermitian finite element new method (QH) in the case of the 0.125-meter cell division is taken as the benchmarking result, the result accuracy in the case of the other cell division can be analyzed. Wherein BMQH-Dif represents the quintic Hermite finite element New method (QH) relative error; BMCH-Dif represents the relative error of the conventional cubic Hermite finite element method (CH); it can be seen that the result of the quintic hermite finite element new method (QH) does not change much regardless of the unit thickness, whereas the conventional cubic hermite finite element method (CH) necessarily requires extremely fine meshing to obtain a more accurate result, and the coarser the mesh, the larger the error. BM _ FDM in table 7 indicates that the calculation is analyzed by using the conventional finite element method (CH), but the post-processing uses a 5-point difference method to obtain information of the second derivative, i.e. curvature, by using the obtained information difference, and then calculates the bending moment value. By combining the above information, the five-dimensional Hermite shape function (finite element) finite element method adopted by the patent is mainly applicable to the following working conditions or examples:
and (3) accurately predicting the numerical finite element of the maximum bending moment/shearing force of various flexible bodies under the action of any load.
The accurate numerical finite element accurate prediction of the high-order natural vibration frequency of various flexible bodies (including flexible bodies which cannot resist bending, such as anchor chains and ropes with extremely small bending rigidity).
When the change gradient of the transverse load born by the flexible body is large, the prediction of the internal force of the flexible body can be obviously improved.
The internal force prediction with higher precision is carried out on the condition that a flexible body with rigidity bears larger bending (the problem of large curvature);
based on the algorithm, the internal force calculated by the flexible body static force and dynamic force problem is more accurate, and the method can be concluded that the method is not only suitable for the flexible body strength problem, but also has remarkable help for the prediction accuracy of the flexible body fatigue problem.
It should be noted that the above description is not intended to limit the present invention, and the present invention is not limited to the above examples, and those skilled in the art may make variations, modifications, additions or substitutions within the spirit and scope of the present invention.
Claims (5)
1. A flexible body structure high-order nonlinear finite element numerical simulation method based on a quintic hermitian function is characterized by comprising the following specific steps:
the method comprises the following steps: based on a Quintic Hermite (Quintic Hermite) high-order interpolation theory, a Quintic Hermite interpolation shape function is deduced, and the Quintic Hermite interpolation shape function comprises 6 Quintic functions:
wherein, the first and the second end of the pipe are connected with each other,
x a and x b Respectively as the start point coordinate and the end point coordinate of the space finite element node, and the coordinate interval range under the general coordinate system x is x E [ x ∈ [ ] a ,x b ](ii) a The coordinate interval range of xi under the local coordinate system is xi epsilon [0,1]；
The attribute equation of the quintic hermitian interpolation shape function is as follows:
step two: determining a flexible body structure unit and parameters to be solved by the flexible body structure unit, determining a physical model of the flexible body structure unit according to size parameters of the flexible body structure unit, and establishing a mathematical motion partial differential equation of the flexible body structure unit in space and time; analyzing to obtain the deformation characteristic of the flexible body structure unit in the axial direction, and obtaining the stretching limit condition of the flexible body structure unit along the axial direction;
the mathematical motion partial differential equation of the flexible body structure unit in space and time is specifically as follows:
wherein, M m Is a quality matrix;
r、
q is a variable which changes with the arc length s and the time t under an arc coordinate system;
wherein r is a space instantaneous position vector of the flexible body structure unit after deformation;
b is the bending rigidity of the flexible body structure unit, B = EI, E is the Young modulus of the material of the flexible body structure unit, and I is the moment of inertia of the flexible body structure unit to the neutral axis;
q is the load distribution of the external transverse load along the length direction of the flexible body structure unit;
the black point at the top of the symbol represents the derivative with respect to time, and the two points represent the second derivative with respect to time; the' in the upper right hand corner of the symbol represents the first derivative to the spatial arc length; the "in the upper right hand corner of the symbol represents the second derivative to the spatial arc length; the "' in the upper right hand corner of the notation represents the third derivative to spatial arc length; the "in the upper right hand corner of the symbol represents the fourth derivative to spatial arc length;
step three: substituting the quintic Hermite interpolation shape function (1) obtained in the first step into a mathematical motion partial differential equation (3) of the flexible body structure unit in space and time, integrating the whole flexible body structure unit by adopting partial integration, and establishing a nonlinear system equation on the space unit of the flexible body structure unit by combining the quintic Hermite interpolation shape function attribute equation (2) in the first step:
wherein L represents the length of the flexible body structural unit;
a i (s) represents the expression of the quintic hermite interpolation shape function in the overall arc coordinate system;
a i (s)＝J i ·φ i (xi) (i =1 to 6), and establishing a conversion relation of the overall coordinate system and the shape function expression under the unit coordinate system;
J i representing Jacobian transformation coefficients under different coordinate systems;
the relationship between subscript i and length L of the flexible body structural unit is as follows:
step four: applying boundary conditions at both ends of the flexible body structure unitEquation (5) with the right side of equal sign is defined as the generalized force F of the flexible body structural unit i Generalized force F of the flexible body structural unit i The expression is as follows:
step five: discretizing the parameters in the nonlinear system equation (5) in the third step by using a tensor algorithm:
obtaining the spatial instantaneous position distribution r (s, t) and the high-order derivative r of the flexible body structural unit through quintic Hermite interpolation shape function simulation (n) (s，t)；
By a quadratic interpolation function p m (s) simulation to obtain effective tension distribution inside the flexible body structure unit
Mass distribution M (s, t) inside the flexible body structure unit and bending rigidity distribution B(s) inside the flexible body structure unit;
the specific simulation physical quantity equation is as follows:
step six: substituting the numerical simulation physical quantity equation (7) obtained in the step five into a nonlinear system equation (5) on a space unit of the flexible body structure unit, and combining the generalized force F of the flexible body structure unit in the step five i And (3) deriving a discrete equation in the flexible body structural unit by using an expression (6) and an attribute equation (2) of a quintic Hermite interpolation shape function:
α ikm an integral coefficient matrix related to a bending stiffness term generated by the Galerkin method;
β ikm an integral coefficient matrix related to a tension term generated by the Galerkin method;
γ ikm an integral coefficient matrix related to an inertial force term generated by the Galerkin method;
q in vectors generated for external laterally distributed loads applied to the flexible body elements;
F in is the generalized force vector at the unit node;
step seven: performing coordinate system transformation on discrete equations (8) within the flexible body structural units: the coordinate system of the quintic Hermite interpolation shape function is processed from the space global coordinate system through the Jacobian transformation coefficient J i Converting to a unit local coordinate system to obtain a discrete equation in a flexible body structure unit under the local coordinate system, and solving a rigidity coefficient matrix in equation (8);
wherein, the discrete system equation (9) in the flexible body structure unit under the local coordinate system is as follows:
all stiffness coefficient matrices in equation (8) are solved for the relationship:
step eight: establishing a stretching limiting condition of the flexible body structure unit along the axial direction in a simultaneous step II, repeating the step III to the step V, discretizing the stretching limiting condition of the flexible body structure unit along the axial direction to obtain a discrete equation (11) corresponding to the stretching limiting condition of the flexible body structure unit along the axial direction;
step nine: and solving a nonlinear system coupling equation on the space unit of the flexible body structure unit by a Newton iteration method according to a discrete system equation (9) in the flexible body structure unit under the local coordinate system and a discrete equation (11) corresponding to the stretching limit condition of the equation flexible body structure unit along the axial direction.
2. The fifth order hermitian function based high order nonlinear finite element numerical simulation method of a flexible body structure according to claim 1, wherein in the second step, the deformation characteristics of the flexible body structure unit in the axial direction at least include a condition that the flexible body structure unit is not stretchable in the axial direction and a condition that the flexible body structure unit is stretched in the axial direction;
for the case that the flexible body structure unit is not stretchable along the axial direction, the axial stretching limiting conditions are as follows:
r'·r'＝1 (4-a)
in the case of deforming and stretching the flexible body structure unit along the axial direction, the axial stretching limiting conditions are as follows:
r'·r'＝(1+ε) 2 (4-b)
in equations (4-a) and (4-b), the middle black dot represents the inner product or dot product of the vectors;
epsilon represents the axial strain of the flexible body structural unit,
wherein T is the local axial tangential tension, A represents the cross-sectional area of the flexible body structure unit, and E is the Young modulus of the flexible body structure unit.
3. The fifth order hermitian function-based numerical simulation of a flexible body structure high order nonlinear finite element according to claim 2, wherein in the eighth step, for the case that the flexible body structure unit is not stretchable along the axial direction, the discrete equation corresponding to the stretching constraint condition along the axial direction is:
for the flexible body structure unit having deformation stretching along the axial direction, the discrete equation corresponding to the stretching limitation condition of the axial direction is:
wherein, the relational expression of flexible body structure unit axial strain and effective pulling force of axial and riser inside and outside hydrostatic pressure does:
A f the cross section area is corresponding to the outer diameter of the flexible body structure unit;
A c the cross section area is corresponding to the inner diameter of the flexible body structure unit;
ρ f gyA f -ρ c gyA c the pressure difference of the static water external pressure and the internal pressure of the external fluid and the internal fluid of the corresponding flexible body structure unit at the vertical coordinate y is obtained;
when the flexible body structure unit is not in the fluid, then rho f gyA f -ρ c gyA c ＝0；
The coefficient matrix in the equation can be solved by the following integral relation:
4. the method of claim 3, wherein the method comprises the steps of:
when the flexible body structure unit is in a statics state, a Newton iteration method is adopted to solve the nonlinear system equation on the space unit of the flexible body structure unit, the letter r represents the number of iteration steps, and the iterative algorithm formulas of the step r and the step r-1 are shown in the formulas (14) and (15) in detail:
5. the fifth order hermitian function based flexible body structure high order nonlinear finite element numerical simulation method of claim 3, wherein:
when the flexible body structure unit is in a dynamic state, performing numerical integration on time by adopting a Newmark-beta numerical calculation method, performing time step division on finite element simulation time, and expressing the time step by using s; for the current s time step, when a Newton iteration method is adopted to solve the nonlinear system equation on the space unit of the flexible body structure unit, the letter r represents the number of iteration steps, and the iterative algorithm formula of the r step and the r-1 step is detailed in formulas (16) and (17):
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