CN108647411A - A kind of truss stress model modeling method expressed based on the spaces Grassmann and recurrence surface - Google Patents
A kind of truss stress model modeling method expressed based on the spaces Grassmann and recurrence surface Download PDFInfo
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Abstract
The invention discloses a kind of truss stress model modeling methods expressed based on the spaces Grassmann and recurrence surface, characterized in that includes the following steps：1）Reasonable L, W curve is redefined on the basis of the spaces Grassmann and Grassmann coordinates, and provides corresponding curve equation；2）Curved surface formula of L, W curved surface on triangle domain is provided on the basis of barycentric coodinates, and L, W curved surface are expanded into triangle domain from the rectangular domain of tensor product form；3）Establish particle truss model.This method can improve the fitting degree of simulation figure and the precision approached, the grid surface truss force analysis model force analysis established on this basis is simple and practicable, accuracy is high, operability is good, the model is not only applicable to the force analysis of rigid objects, in the case where reconnaissance is closeer, also there is ideal effect to the emulation of flexible fabric.
Description
Technical field
Model field the present invention relates to curved surface and curved surface stress model, more particularly to it is a kind of based on the spaces Grassmann and
The truss stress model modeling method of recurrence surface expression.
Background technology
In recent years, the service quality of the authenticity, real-time and the interactivity that are presented to computer graphical with people is more next
Higher, the trend that simultaneous geometry designs object is combined towards diversity, particularity and topological structure complexity is increasingly
Obviously, Computer-aided Geometric Design has obtained apparent development.From the point of view of research field, Computer-aided Geometric Design technology
From traditional research representation of a surface, surface intersection close surface joining, extend to curved surface deformation, curve reestablishing, Surface Simplification and
Curved surface potential difference；From the point of view of representation method, the discrete moulding characterized by grid subdivision is combined with traditional block sequences so that
It is made great progress in the design processing of vivid feature animation and sculptured surface.
Ron Goldman are in the research for the algebraic foundation theory for doing Computer-aided Geometric Design, it is proposed that use
The concept in the spaces Grassmann.He is introduced into the concept of " particle " in classical physics in the spaces Grassmann, make particle and
Vector forms a new vector space together.To provide a kind of novelty for Computer-aided Geometric Design theory and practice
Thought.Ron Goldman are pointed out：In Computer-aided Geometric Design, polynomial curve curved surface and its control point, control are more
Side shape or control net, control array, are generally all located at affine space.But for Rational Curves and Surfaces, affine space is clearly
It is not enough.At this point, the spaces Grassmann just seem very strong.Rational Curves and Surfaces should be regarded as multinomial song
Line curved surface is from Grassmann space to affine space or the projection of projective space, control structure should be by Grassmann
What the particle in space was constituted, rather than be made of the homogeneous point in the ordinary point or projective space in affine space.
Invention content
The purpose of the present invention is in view of the deficiencies of the prior art, and provide a kind of bent based on the spaces Grassmann and recurrence
The truss stress model modeling method of face expression.This method can improve the fitting degree of simulation figure and the precision approached,
The grid surface truss force analysis model force analysis established on the basis of this is simple and practicable, accuracy is high, operability is good, should
Model is not only applicable to the force analysis of rigid objects, in the case where reconnaissance is closeer, also has ideal to the emulation of flexible fabric
Effect.
Realizing the technical solution of the object of the invention is：
Based on the truss stress model modeling method that the spaces Grassmann and recurrence surface are expressed, unlike the prior art
, include the following steps：
1) reasonable L, W curve is redefined on the basis of the spaces Grassmann and Grassmann coordinates, and provides phase
The curve equation answered：
In common affine space, it is as follows that reasonable L-curve common are recursive form：
Reasonable L-curve basic function form is represented by：
In the high one-dimensional spaces Grassmann, reasonable L-curve is hinted obliquely at for polynomial curve, and reasonable L-curve is high one-dimensional
The reference representation in the spaces Grassmann is：
ωiFor weight factor, PiVertex in order to control, vi'For power because
Son.
Reference representation by reasonable L-curve in the high one-dimensional spaces Grassmann is mapped onto affine space, obtains reasonable
L-curve general expression is：When t ∈ [0,1], converted by reasonable L-curve
For reasonable W curves, reasonable W curves have convexity-preserving, and reasonable W curved basic functions form is represented by：
And in the high one-dimensional spaces Grassmann, reasonable W Curvilinear mappings are that multinomial is bent
Line, the case where taking zero of holding power, reference representation of the reasonable W curves in the high one-dimensional spaces Grassmann are：
Reasonable W curves the high one-dimensional spaces Grassmann reference representation projection to affine space, obtain reasonable W
The general expression of curve is：
ωiFor weight factor, PiVertex in order to control, vi'For power because
Son；
2) curved surface formula of L, W curved surface on triangle domain is provided on the basis of barycentric coodinates, by L, W curved surface from tensor
The rectangular domain of product form expands to triangle domain：The expression-form of reasonable L curved surfaces has rectangular domain expression-form and triangle domain table
Up to form, the rectangular domain expression-form of reasonable L curved surfaces is in affine space：
ωi,jFor weight factor,
Pi,jVertex in order to control；
The reference representation of reasonable L curved surfaces is in the high one-dimensional spaces Grassmann：
The reference representation projection of reasonable L curved surfaces obtains affine sky to affine space in the high one-dimensional spaces Grassmann
Between in reasonable L curved surfaces general expression：
ωi,jFor weight factor, Pi,jIn order to control
Vertex, vi',j'For weight factor；
Tri patch expression-form for reasonable L curved surfaces in affine space is：
It is built upon centre coordinate base
The basic function of reasonable L-curve triangle domain on plinth, i+j+k=n, the reasonable L in the high one-dimensional spaces Grassmann
The reference representation of curved surface is：
Work as s, t ∈ [0,
When 1], reasonable L curved surfaces are converted into reasonable W curved surfaces, and the basic function form of reasonable W curved surfaces is represented by affine space:
And in high one-dimensional Grassmann
Space, reasonable W curved surfaces are hinted obliquely at for polynomial surface, reference representation of the reasonable W curved surfaces in the high one-dimensional spaces Grassmann
For：
The reference representation projection in the high one-dimensional spaces Grassmann obtains the general table of reasonable W curved surfaces to affine space
Up to formula：
It is for the representation in affine space on the triangle domain of reasonable W curved surfaces：
The reference representation of reasonable W curved surfaces is in the high one-dimensional spaces Grassmann：
The reference representation projection of reasonable W curved surfaces obtains affine sky to affine space in the high one-dimensional spaces Grassmann
Between in reasonable W curved surfaces general expression：
3) particle truss model is established：The construction that realization method carries out reasonable L, W curved surface is calculated using Blossom, is introduced reasonable
L, W curved surfaces parameterize three-dimensional scattered points, and carry out curve reestablishing, then on this basis, according to minimum energy principle, knot
Physical constraint condition is closed, particle truss model is established, model is finally attributed to mathematical programming problem, it is complete by constrained optimization
The stress of pairs of grid surface is analyzed, for A on grid surface SiThe suffered power of point, is generally divided into internal stressWith it is outer
PowerInternal stressDirection with cut arrow direction it is identical or on the contrary, internal stress resultant forceIn tangent plane, by phase on curved surface
Adjoint point interaction is determined that external force is with joint effortsIt generally can be analyzed to " vertical " power" transverse direction " powerHere so-called
" vertical " be perpendicular to tangent plane, i.e., in tu×tvDirection on；So-called " transverse direction " be in tangent plane, power
Addition and subtraction are applicable in the Vector triangle of vector, A on grid surface SiThe resultant force equation of point： WithWithDirection is vertical, when balance,I.e. vertical force is normal
It is decomposed into the opposite pressure of a pair of aspect equal in magnitudeWith support forceAndI.e. cross force is decomposed into a pair of of thrust Ft iWith
Resistance Whereinη is skin-friction coefficient, so A on grid surface SiThe conjunction of point
Equilibrium equation is：
To all n stress points on grid surface S, then grid surface S resultant forces equilibrium equation is：Because internal stress is mutual between grid surface S consecutive points
Caused by effect, occur in pairs, and direction equal in magnitude is on the contrary, so all n points in grid surface S
The sum of internal stress have：The resultant force equilibrium equation of grid surface S is reduced to as a result,：Grid surface S is for the resultant force moment equation of any point O：
Have when balance：
WhereinIt is to act on AiInternal stress resultant force, F at pointt iIt is to act on AiPoint horizontal thrust resultant force,It is effect
In AiPoint horizontal resistance resultant force,It is to act on AiPoint pressure at right angle resultant force,It is to act on AiPoint vertical support power resultant force,RespectivelyTo the arm of force of O points, Li=OAiIf setting O points as coordinate origin,OA can be acquired by Differential GeometryiA is crossed with grid surface SiThe angle β of the tangent plane of pointi,
According to physical property, pressureAnd support forceDirection equal in magnitude is opposite, vertically and tangent plane；And planted agent
Power resultant forceResistanceWith thrust Ft iAll in tangent plane, therefore：So 3. formula can simplify
For：
If by AiFrom the point of view of at quality be miParticle, then AiGrassmann coordinates (the m of pointi(xi,yi,zi),mi), then matter
Point AiThe physical parameters such as the velocity and acceleration generated by stress are：
In the spaces Grassmann：Understand acceleration
Grassmann coordinates can be converted into (∑ Fi,mi),
According to minimum energy principle, when system resultant force reaches balance, grid surface S inner potentials summation is minimum, then mould
Type is attributed to mathematical programming problem, as convex programming：Wherein Ω is convex in linear space X
Collection, f is the real value convex functional on Ω, G：Ω → Z is Convex Mappings, and Z is the normed property space for having positive convex cone, according to broad sense
Lagrange Multiplier Theorems, then convex programming：Wherein μ0It is limited, then exists in Z*
Lagrange multipliersSo that：In above formula
Referred to as Lagrange functionals, according to broad sense target spot theorem, it is assumed that generalized L agrange functionals L (x, z*) withIt is wide for it
Adopted saddle point, as long as G is Convex Mappings, then x0The as optimal solution of planning problem can be by grid song for most of engineering problems
Face S geometrical model approximations are reduced to the truss model of n particle composition, then
Inner potential summation is：
Gravitional forceIt is combined into：
Kinetic energyIt is combined into：
Truss model gross energy on grid surface is：
Using the truss model gross energy on grid surface as object function, in conjunction with constraints be formula 1., 2. complete net
It can be arbitrary topology knot that the advantages of truss force analysis on lattice curved surface, the truss model of above method structure, which is grid node,
Structure is suitble to Modeling of Complex Surface.
This method can improve the fitting degree of simulation figure and the precision approached, the grid surface established on the basis of secondary
Truss force analysis model force analysis is simple and practicable, accuracy is high, operability is good, which is not only applicable to rigid objects
Force analysis also have ideal effect to the emulation of flexible fabric in the case where reconnaissance is closeer.
Description of the drawings
Fig. 1 is the method flow schematic diagram of embodiment；
Fig. 2-1 is grid surface S stress diagrams in embodiment；
Fig. 2-2 is grid surface in embodiment a little to the torque schematic diagram of O points；
Fig. 3-1 is level bridges structural schematic diagram in embodiment；
Fig. 3-2 is active point load P in embodiment0It is assigned to the schematic diagram of passive point group.
Specific implementation mode
Present disclosure is further elaborated with reference to the accompanying drawings and examples, but is not the limit to the present invention
It is fixed.
Embodiment：
Referring to Fig.1, the truss stress model modeling method expressed based on the spaces Grassmann and recurrence surface, including such as
Lower step：
1) reasonable L, W curve is redefined on the basis of the spaces Grassmann and Grassmann coordinates, and provides phase
The curve equation answered：
In common affine space, it is as follows that reasonable L-curve common are recursive form：
Reasonable L-curve basic function form is represented by：
In the high one-dimensional spaces Grassmann, reasonable L-curve is hinted obliquely at for polynomial curve, and reasonable L-curve is high one-dimensional
The reference representation in the spaces Grassmann is：
ωiFor weight factor, PiVertex in order to control, vi'For power because
Son；
Reference representation by reasonable L-curve in the high one-dimensional spaces Grassmann is mapped onto affine space, obtains reasonable
L-curve general expression is：When t ∈ [0,1], converted by reasonable L-curve
For reasonable W curves, reasonable W curves have convexity-preserving, and reasonable W curved basic functions form is represented by：
And in the high one-dimensional spaces Grassmann, reasonable W Curvilinear mappings are that multinomial is bent
Line, the case where taking zero of holding power, reference representation of the reasonable W curves in the high one-dimensional spaces Grassmann are：
Reasonable W curves the high one-dimensional spaces Grassmann reference representation projection to affine space, obtain reasonable W
The general expression of curve is：
ωiFor weight factor, PiVertex in order to control, vi'For power because
Son；
2) curved surface formula of L, W curved surface on triangle domain is provided on the basis of barycentric coodinates, by L, W curved surface from
The rectangular domain of amount product form expands to triangle domain：The expression-form of reasonable L curved surfaces has rectangular domain expression-form and triangle domain
Expression-form, the rectangular domain expression-form of reasonable L curved surfaces is in affine space：
ωi,jFor weight factor,
Pi,jVertex in order to control, vi',j'For weight factor；
The reference representation of reasonable L curved surfaces is in the high one-dimensional spaces Grassmann：
The reference representation projection of reasonable L curved surfaces obtains affine sky to affine space in the high one-dimensional spaces Grassmann
Between in reasonable L curved surfaces general expression：
ωi,jFor weight factor, Pi,jIn order to control
Vertex, vi',j'For weight factor；
Tri patch expression-form for reasonable L curved surfaces in affine space is：
It is built upon centre coordinate base
The basic function of reasonable L-curve triangle domain on plinth, i+j+k=n, the reasonable L in the high one-dimensional spaces Grassmann
The reference representation of curved surface is：
Work as s, t ∈ [0,
When 1], reasonable L curved surfaces are converted into reasonable W curved surfaces, and the basic function form of reasonable W curved surfaces is represented by affine space:
And in high one-dimensional Grassmann
Space, reasonable W curved surfaces are hinted obliquely at for polynomial surface, reference representation of the reasonable W curved surfaces in the high one-dimensional spaces Grassmann
For：
The reference representation projection in the high one-dimensional spaces Grassmann obtains the general table of reasonable W curved surfaces to affine space
Up to formula：
It is for the representation in affine space on the triangle domain of reasonable W curved surfaces：
The reference representation of reasonable W curved surfaces is in the high one-dimensional spaces Grassmann：
The reference representation projection of reasonable W curved surfaces obtains affine sky to affine space in the high one-dimensional spaces Grassmann
Between in reasonable W curved surfaces general expression：
3) particle truss model is established：The construction that realization method carries out reasonable L, W curved surface is calculated using Blossom, is introduced reasonable
L, W curved surfaces parameterize three-dimensional scattered points, and carry out curve reestablishing, then on this basis, according to minimum energy principle, knot
Physical constraint condition is closed, particle truss model is established：Model is finally attributed to mathematical programming problem, it is right by constrained optimization
The stress of grid surface is analyzed, for A on grid surface SiThe suffered power of point, is generally divided into internal stressAnd external forceInternal stressDirection with cut arrow direction it is identical or on the contrary, internal stress resultant forceIn tangent plane, by adjacent on curved surface
Point interaction is determined that external force is with joint effortsIt generally can be analyzed to " vertical " power" transverse direction " powerHere so-called
" vertical " is perpendicular to tangent plane, i.e., in tu×tvDirection on；So-called " transverse direction " be in tangent plane, the addition of power with subtract
Method is applicable in the Vector triangle of vector, as shown in Fig. 2-1, A on grid surface SiThe resultant force equation of point： WithWithDirection is vertical, when balance,(vertical force) is normal
It is decomposed into the opposite pressure of a pair of aspect equal in magnitudeWith support forceAnd(cross force) is decomposed into a pair of of thrust Ft iWith
Resistance Whereinη is skin-friction coefficient, so A on grid surface SiThe resultant force of point
Equilibrium equation is：
To all n stress points on grid surface S, then grid surface S resultant forces equilibrium equation is：Because of phase interaction of the internal stress between grid surface S consecutive points
With generated, occur in pairs, and direction equal in magnitude is on the contrary, so all n in grid surface S
The sum of the internal stress of point has：The resultant force equilibrium equation of grid surface S is reduced to as a result,：As shown in Fig. 2-2, resultant force moment equations of the grid surface S for any point O
For：Have when balance：
WhereinIt is to act on AiInternal stress resultant force, F at pointt iIt is to act on AiPoint horizontal thrust resultant force,It is effect
In AiPoint horizontal resistance resultant force,It is to act on AiPoint pressure at right angle resultant force,It is to act on AiPoint vertical support power resultant force,RespectivelyTo the arm of force of O points, Li=OAiIf setting O points as coordinate origin,OA can be acquired by Differential GeometryiA is crossed with grid surface SiThe angle β of the tangent plane of pointi,
According to physical property, pressureAnd support forceDirection equal in magnitude is opposite, vertically and tangent plane；And planted agent
Power resultant forceResistanceWith thrust FtI all in tangent plane, therefore：So 3. formula can simplify
For：
If by AiFrom the point of view of at quality be miParticle, then AiThe Grassmann coordinates of point are (mi(xi,yi,zi),mi), then
Particle AiThe physical parameters such as the velocity and acceleration generated by stress are：In the spaces Grassmann：Understand that the Grassmann coordinates of acceleration can be converted into (∑ Fi,mi),
According to minimum energy principle, when system resultant force reaches balance, grid surface S inner potentials summation is minimum, then mould
Type is attributed to mathematical programming problem, as convex programming：Wherein Ω is convex in linear space X
Collection, f is the real value convex functional on Ω, G：Ω → Z is Convex Mappings, and Z is the normed property space for having positive convex cone, according to broad sense
Lagrange Multiplier Theorems, then convex programming：Wherein μ0It is limited, then in Z*Middle presence
Lagrange multipliersSo that：In above formula
Referred to as Lagrange functionals, according to broad sense target spot theorem, it is assumed that generalized L agrange functionals L (x, z*) withIt is wide for it
Adopted saddle point, as long as G is Convex Mappings, then x0The as optimal solution of planning problem can be by grid song for most of engineering problems
Face S geometrical model approximations are reduced to the truss model of n particle composition, then
Inner potential summation is：
Gravitional forceIt is combined into：
Kinetic energyIt is combined into：
Truss model gross energy on grid surface is：
Using the truss model gross energy on grid surface as object function, in conjunction with constraints be formula 1., 2. complete net
Truss force analysis on lattice curved surface.The advantages of truss model is that grid node can be randomly topologically structured, is suitble to complicated bent
Face moulding.
Specifically, on the basis of surface mesh truss model, practical application is converted to the rail loads in steelframe bridge
Project is established for the rail loads analysis model in steelframe bridge, in the design of the crossbeam in steelframe bridge, it is often necessary to
Load is transferred on another point group from a point group, when conversion will not only meet the balance of power and torque, but also calculate knot
Fruit is reasonable, accurate, therefore the load transfer of crossbeam is calculated using the load model on grid.
As shown in figure 3-1, loading analysis is carried out on surface mesh truss model to the beam surface of level bridges, it is assumed that steel
Beam surface upper in bridge formation is on a two dimensional surface, it is known that the coordinate (X, Y) and one of point of two groups of point groups of beam surface
The distribution that loads on group's (tentatively referred to as active point group) is intended to the load P on active point group to be assigned to another point group and (there is no harm in now
Referred to as passive point group), if the load on every bit on active point group is all assigned to the every bit on passive point group, and ensure quilt
Power and equalising torque on dynamic point group, as shown in figure 3-2, it is assumed that the coordinate on any point on active point group is (X0, Y0), it bears
Lotus is P0, the coordinate of passive point group is (Xi, Yi), i=1,2 ... n, liIt is the distance that actively point arrives passive point group, that is,
The lever of torque is long, and target is to calculate distribution by the load P on dynamic meshi：In this lever system, on active point group
Load P0It is to the elastic potential energy generated of putting on each passive point group：The then proof resilience gesture in lever system
Can be：Wherein K is cross sectional moment of inertia, and J is Young's modulus.
According to the equilibrium equation group of power system：
Resettle Lagrange equations：
According to minimum energy principle, i.e.,When, the elastic potential energy of power system
Summation is minimum：
3. 4. simultaneous can obtain formula with formula：AZ=B is 5.
Wherein：Z=(λ1,λ2,λ3)T, B=(P0,P0X0,P0Y0)T
Due on the generally no longer same straight line of the point on passive point group, therefore 4., 5. simultaneous can obtain：
Wherein：
This is arrived, has obtained a little following the sharing of load of every bit on passive point group according to above-mentioned steps in active point group
Every bit on ring active point group carries out sharing of load, may finally obtain the distribution load that passive point group all the points are obtained,
The force analysis model not only meets the balance of power and torque between active point group and passive point group, and meets practical feelings
Condition.
Claims (1)
1. the truss stress model modeling method expressed based on the spaces Grassmann and recurrence surface, characterized in that including as follows
Step：
1) reasonable L, W curve is redefined on the basis of the spaces Grassmann and Grassmann coordinates, and is provided corresponding
Curve equation：
In common affine space, it is as follows that reasonable L-curve common are recursive form：
Reasonable L-curve basic function form is represented by：
In the high one-dimensional spaces Grassmann, reasonable L-curve is hinted obliquely at for polynomial curve, and reasonable L-curve is high one-dimensional
The reference representation in the spaces Grassmann is：
ωiFor weight factor, PiVertex in order to control, vi'For weight factor,
Reference representation by reasonable L-curve in the high one-dimensional spaces Grassmann is mapped onto affine space, obtains reasonable L-curve one
As expression formula be：When t ∈ [0,1], reasonable W is converted by reasonable L-curve
Curve, reasonable W curves have convexity-preserving, and reasonable W curved basic functions form is represented by：
And in the high one-dimensional spaces Grassmann, reasonable W Curvilinear mappings are polynomial curve, are held power
The case where taking zero, reference representation of the reasonable W curves in the high one-dimensional spaces Grassmann are：
Reasonable W curves the high one-dimensional spaces Grassmann reference representation projection to affine space, obtain reasonable W curves
General expression be：
ωiFor weight factor, PiVertex in order to control, vi'For weight factor；
2) curved surface formula of L, W curved surface on triangle domain is provided on the basis of barycentric coodinates, by L, W curved surface from tensor product shape
The rectangular domain of formula expands to triangle domain：The expression-form of reasonable L curved surfaces has rectangular domain expression-form and triangle domain expression shape
Formula, the rectangular domain expression-form of reasonable L curved surfaces is in affine space：
U ∈ [a, b], v ∈ [c, d], ωiFor weight factor, PiVertex in order to control,
The reference representation of reasonable L curved surfaces is in the high one-dimensional spaces Grassmann：
The reference representation projection of reasonable L curved surfaces is obtained to affine space in affine space in the high one-dimensional spaces Grassmann
The general expression of reasonable L curved surfaces：
ωi,jFor weight factor, Pi,jVertex in order to control,
vi',j'For weight factor,
Tri patch expression-form for reasonable L curved surfaces in affine space is：
(0≤s, t, s+t≤1),It is built upon reasonable on the basis of centre coordinate
The basic function of L-curve triangle domain, i+j+k=n, the standard scale of reasonable L curved surfaces in the high one-dimensional spaces Grassmann
It is up to formula：
Work as s, when [0,1] t ∈,
Reasonable L curved surfaces are converted into reasonable W curved surfaces, and the basic function form of reasonable W curved surfaces is represented by affine space:
S ∈ [a, b], t ∈ [c, d] and it is empty in high one-dimensional Grassmann
Between, reasonable W curved surfaces are hinted obliquely at for polynomial surface, and reference representation of the reasonable W curved surfaces in the high one-dimensional spaces Grassmann is：
It is high one-dimensional
The spaces Grassmann reference representation projection to affine space, obtain the general expression of reasonable W curved surfaces：
It is for the representation in affine space on the triangle domain of reasonable W curved surfaces：
The reference representation of reasonable W curved surfaces is in the high one-dimensional spaces Grassmann：
The reference representation projection of reasonable W curved surfaces is obtained to affine space in affine space in the high one-dimensional spaces Grassmann
The general expression of reasonable W curved surfaces：
3) particle truss model is established：The construction that realization method carries out reasonable L, W curved surface is calculated using Blossom, it is bent to introduce reasonable L, W
It is parameterized in face of three-dimensional scattered points, and carries out curve reestablishing, the stress of grid surface is analyzed in completion, for grid song
A on the S of faceiThe suffered power of point, is generally divided into internal stressAnd external forceInternal stressDirection is identical or phase with arrow is cut in direction
Instead, internal stress resultant forceIn tangent plane, determined by consecutive points interaction on curved surface, external force resultant forceCan generally it divide
Solution is " vertical " power" transverse direction " powerIt is so-called it is " vertical " be perpendicular to tangent plane, i.e., in tu×tvDirection on；Institute
" transverse direction " of meaning is in tangent plane, and the addition of power and subtraction are applicable in the Vector triangle of vector, A on grid surface SiThe conjunction of point
Power equation： WithWithDirection is vertical, when balance,
I.e. vertical force is often decomposed into the opposite pressure of a pair of aspect equal in magnitudeWith support forceAndI.e. cross force is decomposed into one
To thrust Ft iWith resistance Whereinη is skin-friction coefficient, so grid surface S
Upper AiPoint resultant force equilibrium equation be：
To all n stress points on grid surface S, then grid surface S resultant forces equilibrium equation is：
Interaction of the internal stress between grid surface S consecutive points
It is generated, occur in pairs, and in grid surface S on the contrary, there there is the sum of the internal stress of all n points in direction equal in magnitude：
The resultant force equilibrium equation of grid surface S is reduced to as a result,：
Grid surface S is for the resultant force moment equation of any point O：
Have when balance：
WhereinIt is to act on AiInternal stress resultant force, F at pointt iIt is to act on AiPoint horizontal thrust resultant force,It is to act on Ai
Point horizontal resistance resultant force,It is to act on AiPoint pressure at right angle resultant force,It is to act on AiPoint vertical support power resultant force,RespectivelyTo the arm of force of O points, Li=OAiIf setting O points as coordinate origin,OA can be acquired by Differential GeometryiA is crossed with grid surface SiThe angle β of the tangent plane of pointi, pressureWith
Support forceDirection equal in magnitude is opposite, vertically and tangent plane；And internal stress resultant forceResistanceWith thrust Ft iAll exist
In tangent plane, therefore：So 3. formula can be reduced to：
If by AiFrom the point of view of at quality be miParticle, then AiThe Grassmann coordinates of point are (mi(xi,yi,zi),mi), then particle Ai
The physical parameters such as the velocity and acceleration generated by stress are：
In the spaces Grassmann：Understand the Grassmann of acceleration
Coordinate can be converted into (∑ Fi,mi),
According to minimum energy principle, when system resultant force reaches balance, grid surface S inner potentials summation is minimum, and then model is returned
It ties in mathematical programming problem, as convex programming：Wherein Ω is the convex set in linear space X, f
It is the real value convex functional on Ω, G：Ω → Z is Convex Mappings, and Z is the normed property space for having positive convex cone, according to generalized L agrange
Multiplier Theorem, then convex programming：Wherein μ0It is limited, then in Z*In there are Lagrange multipliersSo that：In above formulaReferred to as Lagrange
Functional, according to broad sense target spot theorem, it is assumed that generalized L agrange functional L (x, z*) withFor its generalized saddle point, as long as G is
Convex Mappings, then x0Grid surface S geometrical model approximations are reduced to the purlin of n particle composition by the as optimal solution of planning problem
Frame model, then
Inner potential summation is：
Gravitional forceIt is combined into：
Kinetic energyIt is combined into：
Truss model gross energy on grid surface is：
Using the truss model gross energy on grid surface as object function, in conjunction with constraints be formula 1., it is bent 2. to complete grid
Truss force analysis on face.
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