Tomtat luan an tieng anh holehuyphuc 22 09 2020

That not only keeps size of the optimizationproblem small but also ensures the numerical procedure truly mesh-free.One more advantage of iRBF method, which is absent in almost meshlesson

MINISTRY OF EDUCATION AND TRAINING

HO CHI MINH CITY UNIVERSITY OF TECHNOLOGY AND EDUCATION

HO LE HUY PHUC

DEVELOPMENT OF NOVEL MESHLESS METHOD FOR LIMIT AND SHAKEDOWN ANALYSIS OF STRUCTURES & MATERIALS

SUMMARY OF DOCTORAL THESIS

MAJOR: ENGINEERING MECHANICS

Ho Chi Minh city, 3rd August 2020

MINISTRY OF EDUCATION AND TRAINING

HO CHI MINH CITY UNIVERSITY OF TECHNOLOGY AND EDUCATION

HO LE HUY PHUC

DEVELOPMENT OF NOVEL MESHLESS METHOD FOR LIMIT AND SHAKEDOWN ANALYSIS OF STRUCTURES & MATERIALS

MAJOR: ENGINEERING MECHANICS

Supervisors:

1 Assoc Prof Le Van Canh

2 Assoc Prof Phan Duc Hung

Reviewer 1:

Reviewer 2:

Reviewer 3:

The proposed research is essentially concerning on the development ofpowerful numerical methods to deal with practical engineering problems.The direct methods requiring the use of a strong mathematical tool and aproper numerical discretization are considered

The current work primarily focuses on the study of limit and down analysis allowing the rapid access to the requested information ofstructural design without the knowledge of whole loading history For themathematical treatment, the problems are formulated in form of mini-mizing a sum of Euclidean norms which are then cast as suitable conicprogramming depending on the yield criterion, e.g second order cone pro-gramming (SOCP)

shake-In addition, a robust numerical tool also requires an excellent cretization strategy which is capable of providing stable and accurate so-lutions In this study, the so-called integrated radial basis functions-basedmesh-free method (iRBF) is employed to approximate the computationalfields To eliminate numerical instability problems, the stabilized conform-ing nodal integration (SCNI) scheme is also introduced Consequently, allconstrains in resulting problems are directly enforced at scattered nodesusing collocation method That not only keeps size of the optimizationproblem small but also ensures the numerical procedure truly mesh-free.One more advantage of iRBF method, which is absent in almost meshlessones, is that the shape function satisfies Kronecker delta property leadingthe essential boundary conditions to be imposed easily

dis-In summary, the iRBF-based mesh-free method is developed in nation with second order cone programming to provide solutions for directanalysis of structures and materials The most advantage of proposed ap-proach is that the highly accurate solutions can be obtained with lowcomputational efforts The performance of proposed method is justifiedvia the comparison of obtained results and available ones in the literature

well-in the field of limit and shakedown analysis is dealwell-ing with the nonlwell-inearconvex optimization problems From the mathematical point of views, theresulting problems can be solved using different optimization techniquesusing linear or nonlinear algorithms.

In addition, owing to the increasing use of composite and neous materials in engineering, the computation of micro-structures atlimit state becomes attracted in recent years Known as the innovativemicro-mechanics technique, homogenization theory is such an efficient toolfor the prediction of physical behavior of materials The macroscopic prop-erties of heterogeneous materials can be determined by the analysis at themicroscopic scale defined by the representative volume element (RVE).The implementation of limit analysis for this problem is similar to oneformulated for macroscopic structures A number of numerical approachesfor direct analysis of isotropic, orthotropic, or anisotropic micro-structureshave been developed and achieved lots of great accomplishments

heteroge-1.2 Research motivation

Numerical methods are the most efficient tools for current studies inthe field of limit and shakedown analysis As mentioned above, a num-ber of researchers have devoted their effort to develop the robust ap-proaches for this area The numerical procedures using continuous field,semi-continuous field (Krabbenhoft et al [1]), or truly discontinuous field(Smith and Gilbert [2]) have been executed with the support of finite el-ement method (FEM) However, there are several matters of mesh-basedprocedures, which need to be handled, for instance, locking problems, meshdistortion and highly sensitive to the geometry of the original mesh, par-ticularly in the region of stress or displacement singularities In order toimprove the computational aspect of FEM, a number of studies proposedthe adaptive technique for limit and shakedown analysis, the achievementcan be found in the works of Christiansen and Pedersen [3], Borges et al.[4], Franco et al [5], Lyamin and Sloan [6], Cecot [7], Ngo and Tin-Loi [8],Ciria et al [9], Le [10] However, the whole process is complicate and re-quires the fine meshing to obtain the expected results An improving form

of FEM named SFEM (smoothed finite element method) is also applied inworks of Le et al [11, 12], Tran et al [13], Nguyen-Xuan et al [14] Gener-ally, SFEM is better than FEM in terms of stability and convergence, butthis method does not surmount all disadvantages of FEM caused by themesh Recently, mesh-free methods are also extended to direct analysis.Among them, Element-free Galerkin (EFG) method is the most interestedchoice, several typical studies can be noted here as Chen et al [15, 16],

Le et al [17–20] Besides, some other meshless procedures have been alsosuccessfully applied to this area such as Natural Element method (NEM

- Zhou et al [21, 22]), Radial Point Interpolation method (RPIM - Liuand Zhao [23]) In comparison with the traditional approaches, mesh-freemethods possess the high-order shape function, hence above disadvantagescan be overcame However, it should be noted that several meshless meth-ods lack Kronecker-delta property leading to the difficulty in imposing theessential boundary conditions Owing to the advantages of shape function

as mentioned in previous sections, iRBF method can provide an efficienttreatment for those obstacles arising in whole process of formulating andsolving optimization problems According to the author’s knowledge, theapplications of iRBF method are focused on the fields of solving PDEs [24–27], fluid mechanics [28], or elastic analysis of solid and fracture mechanics[29] The development of iRBF method for limit and shakedown analysiswill be a new contribution to this area In addition, in previous studies us-ing iRBF, the numerical integration is carried out utilizing Gauss points,

increasing the computational cost Therefore, the stabilized approximationbased on the combination of iRBF approximation and SCNI will improvethe computational aspect of proposed numerical method.

Moreover, solving limit and shakedown problem requires to handlethe optimization problem involving either linear or non-linear constrains.The traditional way to overcome this drawback is linearizing non-linearconvex yield criteria The efficient tools, for instance, Simplex algorithm(Anderheggen and Knopfel [30], Christiansen [31]), can be used However,

a large number of constrains and variables in the optimization problemsare required to obtain the sufficiently accuracy results, which increase thecomputational cost On one other hand, that is the attempts to deal withthe convex yield criteria using non-linear packages Although the highly ac-curate solutions can be obtained, the expensive cost is the major trouble

of this scheme In framework of limit analysis, the primal-dual point algorithm (Christiansen and Kortanek [32], Andersen and Chris-tiansen [33]) is well-known as one of most robust and efficient algorithms inhandling the optimization problems with large-scale nonlinear constrains.Therefore, extending of this scheme to the shakedown formulation will lead

interior-to more advantages for direct analysis of either structures or materials.Besides, the earliest application of direct analysis for microscopic struc-tures can be found in studies of Buhan and Taliercio [34], Taliercio [35],Taliercio and Sagramoso [36], where the limit load of typical problems weredetermined The homogenization theory was applied to limit analysis us-ing linear programming in works of Francescato and Pastor [37], Zhang et

al [38], Weichert et al [39, 40], Chen et al [41] Besides that, the ear programming were also employed for direct analysis of heterogeneousmaterials by Carvelli et al [42], Li et al [43–47], Hachemi et al [48], Le

nonlin-et al [49] Actually, almost studies dealt with the isotropic or anisotropicmaterials using linear or nonlinear programming with the support of fi-nite element method, the application of mesh-free method in framework ofcomputational homogenization analysis of materials at limit state is stillunavailable

In conclusion, it can be observed that many challenges still remain indeveloping a robust tool to improve the computational aspect of limit andshakedown analysis for structures and materials Present study focuses onthe combination of a discretization scheme and an optimization program-ming to propose an efficient numerical approach for direct analysis method,i.e., (i) the stabilized iRBF mesh-free method will be developed; (ii) theoptimization problems will be formulated using the so-called second-ordercone programming (SOCP) to deal with the convex yield criterion; (iii) the

numerical approach will be applied to handle the direct analysis problemsfor structures and materials.

1.3 The objectives and scope of thesis

The major objective of thesis is developing the integrated radial basisfunctions-based mesh-free method (iRBF method) and the optimizationalgorithm based on conic programming, then extending the numerical ap-proach to limit and shakedown analysis of structures and materials Inorder to obtain above mentioned aims, the following tasks will be carriedout

• Develop the mesh-free method based on integrated radial basis tions and the stability conforming nodal integration (SCNI)

func-• Formulate the kinematic and static formulations of limit and down analysis for structures and materials, then cast as second-ordercone programming

shake-• Solve the resulting optimization problems using highly efficient tools,then compare obtained solutions with those in other studies to esti-mate the computational aspect of proposed approaches

It is important to note that, within the scope of the thesis, proposed merical method will be employed to deal with several common engineeringstructures, such as continuous beam, simple frame, plates, reinforced con-crete slabs, or computational homogenization analysis of micro-structures.The material model is assumed as rigid-perfectly plastic or elastic-perfectlyplastic The 2D and 3D structures are considered under both constant andvariable loads, corresponding to limit and shakedown analysis, respectively.The benchmark problems will be investigated for the comparison purpose;thereby, the computational aspect of proposed approach is evaluated

two-Theorem 1 Upper bound theorem of shakedown analysis

1 Shakedown may happen if the following inequality is satisfied

2 Shakedown cannot happen when the following inequality holds

Theorem 2 Lower bound theorem of shakedown analysis

1 Shakedown occurs if there exists a permanent residual stress field ρ,

statically admissible, such that

ψ λσ E (x, t) + ρ(x) < 0 (2.5)

2 Shakedown will not occur if no ρ exists such that

ψ λσ E (x, t) + ρ(x) ≤ 0 (2.6)The shakedown problem can be considered as maximizing a nonlinearoptimization problem

for shakedown analysis can be reformulated as

1 Upper bound shakedown analysis

It is important to note that when there is only one loading point, i.e.,

m = 1, shakedown formulations will be reduced to a limit analysis problem.

The lower-bound limit analysis can be expressed as

The micro-scale problem can be treated as the boundary value one

in solid mechanics, where the overall strain E are transferred to

micro-structure in form of kinematic boundary constrains At microscopic scale,the local fields is decomposed into two parts: mean term and fluctuationterm The local displacement, strain and stress are now given by

u(x) = E.X + ˜u(x) (2.17a)

where X is the positional matrix of each material point in the tational domain, Σ is the overall stress; ˜ u(x), ˜(x) and ˜ σ(x) denote the

compu-fluctuation parts of displacement, strain and stress rate

For the purpose of enforcing the boundary condition, this study usesthe periodic procedure, where there are the periodicity of fluctuation dis-placement field and anti-periodicity of traction field on RVE boundary

˜

u(x+) = ˜u(x−), on Γu (2.18a)

t(x+) = −t(x−), on Γt (2.18b)where ˜u(x+) and ˜u(x−) are the fluctuation displacement field, t(x+) and

t(x−) are the traction field of positive and negative boundaries, tively

respec-The macroscopic quantities can be calculated from the microscopic

Heterogeneous material Homogenized material

herein, |Ω| denotes the area of RVE

In direct analysis, the principle of macroscopic virtual work can beexpressed as

It is interesting to note that the numerical implementation of limitand shakedown analysis for materials is carried out similarly to those forstructures

2.4 The iRBF mesh-free method

The smooth function u(x) can be approximated based on a given set

of N scattered nodes and the iRBF method as

is in the strategy to construct the shape function clarified following

In this thesis, the RBF functions will be employed to construct thesecond-order derivative of shape function, then the first-order and the orig-inal functions will be calculated using the integrals as

where C1 and C2 are the integral constants; n1 and n2 represent number

of integral constants (n2= 2n1); a is the vector consisting the unknowns.

This thesis uses the multiquadric (MQ) basis function well-known asthe best iRBF function in terms of accuracy

g I(x) =

q

r2

I (x) + (α s d I)2 (2.23)

where r I (x) is the radius of node I and other ones in its influent domain; d I

is the minimum distance measured form node I to its neighbours; α s > 0

is the dimensionless factor used to control the shape parameter α s d I

Estimating the function at the set of N scattered points, Equation

(2.22c) can be rewritten in terms of matrix form as

As a result, vector a can be expressed via the nodal values u =

[u1u2 u N] as

a = R−10 u = ˆ ΨIku (2.25)

Substituting a into Equations (2.22a) - (2.22c), the approximate

func-tion and its derivatives are recalculated as

u h(x) = R0(x) ˆ ΨIku = Φu (2.26a)

u h ,α(x) = R1(x) ˆ ΨIku = Φ,αu (2.26b)

u h ,αβ(x) = R2(x) ˆ ΨIku = Φ,αβu (2.26c)where Φs , Φ s,α , Φ s,αβ denotes the shape function and its derivatives.The iRBF method overcomes an important obstacle of almost meshless

(a) ΦI(x) (b) ΦI,x(x) (c) ΦI,y(x)

(d) ΦI,xx(x) (e) ΦI,yy(x) (f) ΦI,xy(x)

Figure 2.3: The iRBF shape function and its derivatives

methods in enforcing the essential boundary condition caused by the lack

of Kronecker-delta property

In this thesis, a weak form using the stability conforming nodal gration (SCNI) technique [51] is employed Each node will has a integrationarea so-called representative domain ΩJ The smoothed version of strainrate at node ˜ h ij(xJ) can be calculated by

where ΓJ is the boundary of the representative domain; u i and u j are

the displacement components; n is the outward normal of edges bounding

domain ΩJ as Figure 2.4

In the numerical implementation, the smoothed strain ˜(x J) can beexpressed via the compatible condition and the smoothed derivatives of

Figure 2.4: The SCNI technique in a representative domain

shape function is used

Similarly, the smoothed version of second-order derivatives of shapefunction can be calculated from the first-order ones as

with ΦI,α(x) and ΦI,β(x) are the first-order derivatives of shape function

ΦI (x) relating to variables α and β.

Chapter 3

Displacement and equilibrium mesh-free formulation based on integrated radial basis

functions for dual yield design

This chapter presents an application of iRBF-based mesh-free methodfor plane structures at limit state using both of kinematic and static for-mulations

3.1 Numerical examples

This example is the classical punch problem presented in [52], as shown

in Figure 3.1 Due to symmetry, a rectangular region of dimensions B =

5 and H = 2 is considered Appropriate displacement and stress boundary

conditions are enforced as shown in Figure 3.2 For a load of 2τ0, the

analytical limit multiplier is λ = 2 + π = 5.142.

2τ₀

B

a a

B

Figure 3.1: Prandtl problem

Approximations of upper and lower bounds on the actual limit loadfor both dRBF and iRBF methods with various nodal discretizations areexamined Convergence analysis and relative errors in collapse multipliersversus number of variables are also shown in Figures 3.3(a) and 3.3(b),respectively It should be stressed that the mean values of upper and lowerapproximations obtained using the iRBF-based numerical procedures are

in excellent agreement with the analytical solutions for all nodal tions, as shown in Figure 3.3, with less than 0.4% even for coarse nodaldistribution Moreover, the iRBF method used in combination with directnodal integration can remove the volumetric locking behavior and also

B

2 τ₀ a

(a) Computational domain

Table 3.1: Prandtl problem: comparison with previous solutions

Makrodimopoulos and Martin [53] Kinematic, static 5.148 5.141

UB: upper bound, LB: lower bound

result in stable and accurate solutions

In Table 3.1 the solutions obtained using the present methods with

2560 nodes are compared with those in other studies In general, thepresent solutions are close to results in the literature

iRBF (upper bound)

iRBF (mean value)

Analytical solution

dRBF (upper bound) dRBF (mean value)

(a) Bounds on the collapse multiplier

Number of variables

0 5 10 15 20 25 30 35 40 45

iRBF (upper bound) dRBF (upper bound)

(b) Relative error in collapse ers

multipli-Figure 3.3: Bounds on the collapse multiplier versus the number of nodesand variables

3.2 Conclusions

The present contribution has presented displacement and equilibriummesh-free formulation based on integrated radial basis functions (iRBF)for dual yield design problems In the kinematic formulation, the high-order approximation of the displacement fields using the integrated radialbasis functions can prevent volumetric locking Moreover, direct nodal in-tegration of the iRBF approximation not only results in inexpensive com-putational cost, but also overcomes the instability problems In the staticformulation, with the use of iRBF approximation of the stress fields incombination with the collocation method, equilibrium equations and yieldconditions only need to be enforced at the nodes, leading to the reduc-tion in computational effort It has been shown in several examples thatthe mean values of the iRBF upper and lower bounds are accurate, andcan be considered as the actual collapse load multiplier for most practicalengineering problems, for which exact solution is unknown

Chapter 4

Limit state analysis of reinforced concrete slabs using an integrated radial basis function based mesh-free method

This chapter presents an application of iRBF method for upper boundlimit analysis of structures This study aims to estimate the limit load aswell as collapse mechanics of reinforced concrete slabs

4.1 Numerical examples

Rectangular slabs with either simply supported (SSSS) or clamped(CCCC), (·) corresponds to left, bottom, right and top edges respectively,boundary conditions on all edges are considered first It is assumed that

the slabs are isotropic with positive and negative yield moments(m+p =

m−p = m p) in both directions Due to the symmetry, only the upper-rightquarter of plate is modeled, as shown in Figure 4.1

q x

x y

Figure 4.1: Rectangular slab: geometry, loading, boundary conditions andnodal discretization

The influence of the shape parameter α son the limit load factor of asimply supported square slab is studied first The relationship between the

(a) Normalized limit load factor λ+

ver-sus the parameter α s

Number of nodes

0 10 20 30 40 50 60 70

CCCC CCCF CFCF SSSS FCCC FCFC

(b) Limit load factors with different

boundary conditions (b/a = 2)

Figure 4.2: Rectangular slabs: limit load factors

Table 4.1: Rectangular slabs with various ratios b/a: limit load factors

Table 4.2: Results of simply supported and clamped square slabs

e: relative errors; t: CPU-Time; N var: number of variables

computed limit load factors and the parameter α s, is illustrated in Figure4.2(a) It can be seen that for all nodal distribution solutions obtained

when setting α = 2 are better.

Various ratios of b/a are investigated, and Table 4.1 summarizes

com-puted numerical solutions using a regular nodal distribution of 35 × 35

(a) CCCC (b) CCCF (c) FCCC

Figure 4.3: Rectangular slabs (b = 2a) with various boundary conditions:

plastic dissipation distribution

Table 4.3: Square slabs: limit load multipliers in comparison with othermethods

prob-load factors and convergence analysis for the case when b/a = 2 are

illus-trated in Figure 4.2(b)

To illustrate the performance of the iRBF based limit state analysisprocedure, the present solutions and associated computational aspects,including relative errors and number of variables, for simply supported

and clamped square plates (a = b = L) are compared with those obtained

using the CS-HCT [59] and EFG methods [57], see Table 4.2 In Table 4.3,the iRBF solutions are also compared with previously published upper

and lower bounds using displacement discontinuous finite elements [60],equilibrium mesh-free method [18] and equilibrium finite elements [61, 62].

It is evident that these solutions are, in general, in good agreement Theyield patterns in terms of plastic dissipation distribution for rectangularslabs with different boundary conditions are also plotted in Figure 4.3

4.2 Conclusions

A novel rotation-free mesh-free method based on integrated radial basisfunctions has been developed for limit state analysis of reinforced concreteslabs The transverse velocity is approximated without using rotationaldegrees of freedom, and therefore the total number of variables in theresultant optimization problem is kept to a minimum, i.e., equal to thenumber of discretized nodes in the problem domain The proposed for-mulation is tested by applying it to various Nielsen’s reinforced concreteslabs of arbitrary geometries It has been demonstrated that the presentmethod, consisting of high-order shape functions obtained by integratingradial basis functions, can provide accurate collapse load multipliers More-over, the high-order and smooth iRBF approximation is capable of captur-ing yield patterns of arbitrary geometric slabs The present optimizationstrategy based on conic programming enables solutions of practical sizedreinforced concrete slabs to be obtained rapidly It should be noted that inthe mesh-free based numerical procedures for limit state analysis of struc-tures nodes may be moved, discarded or introduced conveniently Hencethe implementation of an h-adaptive scheme is facilitated, and will be thesubject of future research

L/2

(b) Computational modelFigure 5.1: Square plate with a central circular hole: geometry (thickness

t = 0.4R), loading and computational domain

The plate is solved employing only upper-right quarter, see 5.1, and