CN106960078A - A kind of efficient computational methods for being used to solve local material nonlinear problem - Google Patents

Abstract

The invention discloses a kind of computational methods of Efficient Solution local material nonlinear problem, belong to structural analysis technique field.Nonlinear deformation is changed into the virtual load acted on elastic construction by this method, so that stiffness matrix keeps constant in nonlinear analysis, it is only necessary to once decomposed；The partial structurtes information into nonlinear deformation state is only considered in equation building process, the small-scale matrix for representing local nonlinearity to one is then only needed to carry out decomposition operation during iterative, the real-time update in conventional method to extensive Bulk stiffness matrix and decomposition are avoided, the computational efficiency of local nonlinearity problem is effectively improved.In addition, the present invention is not limited to specific material constitutive relation and numerical model, it may be used on numerous subjects and research field for being related to structural nonlinear calculating, with stronger versatility and wide applicability.

Description

A kind of efficient computational methods for being used to solve local material nonlinear problem

Technical field

The invention belongs to structural analysis technique field, in particular, it is related to a kind of non-linear for solving local material The efficient computational methods of problem.

Background technology

Using non-linear behaviour of the technology of numerical simulation computation structure under extreme environment load action in numerous researchs Field is widely used.Due to the non-linear attributes of structural load-deformation relationship, step-type increment need to be generally used Form is simultaneously solved with reference to appropriate iterative algorithm, during this, and the Bulk stiffness matrix of structure need to be in calculating process Constantly it is updated and decomposition operation, with the increase of computation model scale and the degree that becomes more meticulous, the scale of stiffness matrix is also fast Speed increase, now, the real-time update of extensive stiffness matrix and the key for being decomposed into restriction nonlinear problem computational efficiency.It is right In material non-linearity question, generally only nonlinear deformation occurs for structure partial, and remaining major part will be in initial elasticity all the time It is not intended that this feature in deformation state, conventional method calculating process, thus elasticity and the difference of non-linear partial can not be realized It Hua not treat, therefore, how to make full use of the local feature of nonlinear deformation to realize such issues that efficient numerical solution turns into Research emphasis.

The content of the invention

In order to overcome the shortcomings of above-mentioned existing method, the present invention utilizes Woodbury formula, by considering material nonlinearity A kind of local feature of problem, it is proposed that efficient computational methods for structure partial nonlinear problem, calculating process is only needed pair One small-scale matrix for representing local nonlinearity deformation is updated and decomposition operation, it is to avoid to extensive in conventional method The corresponding computing that Bulk stiffness matrix is carried out.Invention has relatively broad applicability, available for the non-linear of various fields In structural analysis.

Technical scheme：

A kind of efficient computational methods for being used to solve local material nonlinear problem, step is as follows：

(1) numerical model for structural analysis is set up, and presets some nonlinear nodes in the structure；

(2) integrally-built initial elasticity stiffness matrix K is calculated_eAnd triangle decomposition is carried out to it, if structure total displacement is certainly N by the number of degrees, then matrix K_eScale be n × n；

(3) walked for any non-linear incremental computations, under increment load Δ F effects, only consider to enter nonlinear deformation Nonlinear node in state regional area and the nonlinear deformation vector for setting up corresponding incremental form：

ΔΕ″_p=[Δ ε "₁ Δε″₂ ... Δε″_m]^T (1)

In formula, m be the incremental step in enter nonlinear deformation state regional area in nonlinear node number；Δε″_i The increment nonlinear deformation of wherein i-th nonlinear node is represented, wherein 1≤i≤m；If each nonlinear node is non-linear Deformation is made up of p component, then Δ Ε "_pIt is vectorial for a pm rank, wherein each element represents a Nonlinear Free degree；

(4) the displacement structure increment under increment load Δ F effects is solved, i.e.,：

In formula, Δ X represents displacement increment；ΔF_p=K '_rΔΕ″_p, it is the nonlinear additional virtual load of representative structure, its Middle matrix K '_rFor the vectorial Δ Ε " of nonlinear deformation_pWith additional virtual load Δ F_pThe stiffness matrix of relation between, its exponent number is n×pm.The vectorial Δ Ε " of nonlinear deformation under given load increment Δ F effects_pFollowing formula can be used to solve：

In formula,It is K '_rTransposition, be pm × n rank matrixes, represent in the elasticity at displacement of joint and nonlinear node Relation between power, elastic internal force refer to the structure with initial elasticity attribute be subjected to displacement change non-linear hour node at produce should Power；K″_prThe relation between the nonlinear deformation at nonlinear node and elastic internal force is represented, is pm × pm rank matrixes, needs basis Material constitutive relation at nonlinear node is determined.

In step 1, nonlinear node position and quantity in structure need to be determined according to the required precision of particular problem, non-thread Property deformation field can be obtained by interpolation method.

In step 3, the nonlinear deformation under free position is equal to the remaining change after being unloaded according to material initial elasticity rigidity Shape；Vectorial Δ Ε "_pIn the nonlinear node deformations of all entrance nonlinear deformation states in lower structure is walked comprising current delta, But do not include the nonlinear node in initial elasticity deformation state to deform.

In step 3, matrix K '_rAndOccur that position is relevant with non-linear in numerical model, and it is non-with material Linear degree is unrelated, and each element in matrix only needs to calculate once, without being computed repeatedly during nonlinear iteration.

In step 3 and step 4, stiffness matrix K_eInverse matrix with vector or multiplication of matrices computing can be by K_eThree Angle is decomposed and back substitution is calculated and completed, and because the matrix keeps constant during analysis, its triangle decomposition only needs to enter before analysis Row is once.

In step 4, the displacement increment accounting equation and Woodbury formula of any non-linear incremental computations step are of equal value, writeable For the mathematic(al) representation of following form：

In the present invention, each step that calculates is to matrixAmount of calculation needed for decomposing turns into using solution When main calculating consumption, the scale of the matrix is pm × pm (suitable with Nonlinear Free number of degrees mesh), represents and enters non-linear The regional area of deformation state.When structure only small part enter it is non-linear when, have pm<<N, now to matrixAmount of calculation consumed in being decomposed will be carried out in far smaller than conventional method to Bulk stiffness matrix Amount of calculation needed for decomposing, so as to be obviously improved the computational efficiency of nonlinear problem.

Beneficial effects of the present invention：

1. efficient NONLINEAR CALCULATION.For the non-linear material non-linearity question for only resulting from structure partial, the present invention Only need to represent the small-scale carry out decomposition operation of local nonlinearity to one when solving, it is to avoid to larger in existing method Whole the stiffness matrix real-time update and decomposition operation that carry out, can effectively be lifted on the premise of necessary computational accuracy is ensured non- The solution efficiency of linear problem.

2. the extensive scope of application.The present invention is not limited to specific material constitutive relation and numerical model, may be used on Numerous subjects and research field for being related to structural nonlinear calculating, thus with wide applicability.

Brief description of the drawings

Fig. 1 is the analysis model figure of embodiment.

Fig. 2 is calculation flow chart of the invention.

Specific embodiment

It is used to solve the efficient numerical computational methods of local material nonlinear problem the invention provides a kind of, below in conjunction with Attached Example and technical scheme, further illustrate the embodiment of the present invention.Drawings and Examples are only to illustrate the invention Embodiment, do not constitute any limitation of the invention.

Embodiment：

1. by taking the numerical computations that some is made up of two node truss elements shown in Fig. 1 as an example, illustrate side proposed by the invention Embodiment of the method in NONLINEAR CALCULATION.For plane stress element, it is considered to its section axial direction strain stress and axle power N non-thread Sexual intercourse, a nonlinear node is set in unit and makes it be located in the middle of unit, as shown in figure 1, point of its section deformation Solving expression formula is：

ε=ε '+ε " (1)

In formula, ε ' represents section line elastic strain, i.e., section when axle power N is loaded onto using section initial elasticity rigidity is become Shape；ε " represents section nonlinear strain.

2. the integrally-built initial elasticity stiffness matrix K of synthesis_eAnd triangle decomposition is carried out to it, i.e.,：

K_e=LDL^T (2)

In formula, L is unit lower triangular matrix, and D is diagonal matrix.

3. the iterative calculation flow of the invention in nonlinear solution processes provided with reference to Fig. 2, any iteration incremental step Calculating can carry out as follows：

Step one：The determination of location mode is carried out, for calculating the nonlinear node in each unit under the conditions of current displacement The materials behavior at place, including force on cross-section N and material tangent modulus E_t, and further whether judging unit enters nonlinear deformation State, if having E at certain unit nonlinear node_t≠E_e, then judge that the unit enters nonlinear deformation state, wherein E_eFor the list The initial elastic modulus of material at first nonlinear node.

Step 2：According to the materials behavior by nonlinear node in each unit obtained by step one, update matrix K '_rWithAnd calculating matrix K "_prIf have m unit (in this example, has m≤4), then matrix into nonlinear deformation state K″_prIt is represented by

Wherein, A is area of section；L is element length；(.)_iRepresent the material properties related to i-th of nonlinear node.

Step 3：Displacement increment Δ X under being acted on using Woodbury formula computation structure increment load Δs F, i.e.,：

The matrix K provided using formula (2)_eTriangle decomposition form, then above formula be related to and matrixRelevant multiplication Computing can be calculated by back substitution and completed.

Step 4：Convergence judges and updated for next load increment for calculating step.Now need to check that gained calculates knot Whether fruit is reached within defined error range, if convergence, carries out the calculating of next incremental step, if not restraining, continue It is iterated.

Claims (1)

1. a kind of efficient computational methods for being used to solve local material nonlinear problem, it is characterised in that step is as follows：

(1) numerical model for structural analysis is set up, and presets some nonlinear nodes in the structure；

(2) integrally-built initial elasticity stiffness matrix K is calculated_eAnd triangle decomposition is carried out to it, if structure total displacement number of degrees of freedom, N, then matrix K_eScale be n × n；

(3) walked for any non-linear incremental computations, under increment load Δ F effects, only consider to enter nonlinear deformation state Nonlinear node in regional area and the nonlinear deformation vector for setting up corresponding incremental form：

ΔΕ″_p=[Δ ε "₁ Δε″₂ ... Δε″_m]^T (1)

In formula, m be the incremental step in enter nonlinear deformation state regional area in nonlinear node number；Δε″_iRepresent it In i-th of nonlinear node increment nonlinear deformation, wherein 1≤i≤m；If the nonlinear deformation of each nonlinear node is by p Individual component is constituted, then Δ Ε "_pIt is vectorial for a pm rank, wherein each element represents a Nonlinear Free degree；

(4) the displacement structure increment under increment load Δ F effects is solved, i.e.,：

$\mathrm{\&Delta;}X=K_e^{-1}(\mathrm{\&Delta;}F+{\mathrm{\&Delta;F}}_p)---\left(2\right)$ In formula, Δ X represents displacement increment；ΔF_p =K '_rΔΕ″_p, be the nonlinear additional virtual load of representative structure, wherein matrix K '_rFor the vectorial Δ Ε " of nonlinear deformation_pWith Additional virtual load Δ F_pThe stiffness matrix of relation between, its exponent number is n × pm；It is non-under given load increment Δ F effects Linear deformation vector Δ Ε "_pFollowing formula can be used to solve：

${\mathrm{\&Delta;E}}_p^{\&prime;\&prime;}={(K_{pr}^{\&prime;\&prime;}-K_r^{\&prime;T}{K}_e^{-1}{K}_r^{\&prime;})}^{-1}{K}_r^{\&prime;T}{K}_e^{-1}\mathrm{\&Delta;}F---\left(3\right)$

In formula,It is K '_rTransposition, be pm × n rank matrixes, represent between the elastic internal force at displacement of joint and nonlinear node Relation, elastic internal force refers to the structure with initial elasticity attribute and is subjected to displacement the stress for changing and being produced at non-linear hour node；K″_pr The relation between the nonlinear deformation at nonlinear node and elastic internal force is represented, is pm × pm rank matrixes, need to be according to non-linear Material constitutive relation at node is determined.