Upper bound limit analysis of plates using a rotation-free isogeometric approach
© Nguyen-Xuan et al.; licensee Springer. 2014
Received: 6 February 2014
Accepted: 16 July 2014
Published: 27 August 2014
This paper presents a simple and effective formulation based on a rotation-free isogeometric approach for the assessment of collapse limit loads of plastic thin plates in bending.
The formulation relies on the kinematic (or upper bound) theorem and namely B-splines or non-uniform rational B-splines (NURBS), resulting in both exactly geometric representation and high-order approximations. Only one deflection variable (without rotational degrees of freedom) is used for each control point. This allows us to design the resulting optimization problem with a minimum size that is very useful to solve large-scale plate problems. The optimization formulation of limit analysis is transformed into the form of a second-order cone programming problem so that it can be solved using highly efficient interior-point solvers.
Results and conclusions
Several numerical examples are given to demonstrate reliability and effectiveness of the present method in comparison with other published methods.
Accurate prediction of the load bearing capacity of plate structures plays an important role in many practical engineering problems. Traditional elastic designs cannot evaluate the actual load carrying capacity of plates and incremental elasto-plastic analyses can become cumbersome and present convergence issues for large-scale structures. Therefore, various limit analysis approaches have been devised to investigate the behavior of structures in the plastic region. Nowadays, limit analysis has become a well-known tool for assessing the safety load factor of engineering structures as an efficient direct method. Due to limitation of analytical methods, various numerical approaches such as finite element methods (FEM) –, meshfree methods ,, and natural element method , just to mention a few, have therefore been developed.
It is also worth adding that mathematical programming is the other key issue in numerical assessment of limit analysis problem. Discrete upper bound limit analysis results in a minimization problem involving linear or nonlinear programming. Linear programming problems can be applied for piecewise linearization of yield criteria, but an important number of additional variables is often needed. However, most of the yield criteria for plates can be formed as an intersection of cones for which the limit analysis problem can be solved efficiently by the primal-dual interior point method , implemented in the MOSEK software package . This algorithm was proved to be a very effective optimization tool for the limit analysis of structures ,,,–, and therefore it will be used in our study.
Isogeometric approach (IGA) has been recently proposed by Hughes et al.  to unify the fields of Computer Aided Design (CAD) and Finite Element Analysis (FEA). The basic idea is that the IGA uses the same basis functions, namely B-splines or non-uniform rational B-splines (NURBS), to describe precisely the geometry, especially containing conic sections and to construct the finite approximation for analysis. It is well known that NURBS functions provide a flexible way to make refinement, de-refinement and degree elevation . They enable to easily achieve continuity up to C(p−1), instead of C0-continuity as it typically happens with traditional FEM. Hence, IGA naturally verifies the C1-continuity of thin plates, which is interested in this study. The IGA has been well known and widely applied to various practical problems – and so on.
Among various plate theories , the classical plate theory (CPT) and the first-order shear deformation plate theory (FSDT) have been widely used in many numerical methods, especially finite elements. The first-order shear deformation plate theory assumes that transverse shear stresses are constants through the thickness and a shear correction factor (SCF) is needed to take into account the non-linear distribution of shear stresses. It is known in FSDT models that the FE approximation functions only require C0-continuity across element boundaries. Such a construction is simple but leads to shear locking problems. In CPT, C1-continuity of approximation fields across element boundaries is needed. Unfortunately, it is difficult to construct FE formulations with C1-continuous approximation. Traditionally, the conforming FE approximation of the Kirchhoff plate model has in general 3 degrees of freedom per node. This is due to the continuity of the rotation solutions. It is also well known in the literature that non-conforming finite element models enable us to relax strict requirements of the continuity of the rotations. Attempts to eliminate the rotational degrees of freedom help us to reduce significantly the total number of degrees of freedom of problem without loss of accuracy of solution. As a result, such approaches promise more benefit for solving large-scale industrial problems ,,. For example, an efficient way of the rotation-free FE approaches for plate and shell analysis is to use C0 basis functions via the so-called cell-centred and cell-vertex finite volume schemes –. The rotation-free isogeometric approach recently proposed is regarded as an alternative way for solving practical problems. The method is conformable to the thin plate/shell theory and the C1-continuity is easily achieved using NURBS basis functions . Several investigations on the rotation-free formulation can be found in the literature, e.g., Bernoulli-Euler beams , Poisson-Kirchhoff plates , multi-patch Kirchhoff-Love shells  and large deformation analysis with rotation-free . It was demonstrated in the aforementioned references that the rotation-free isogeometric approach is a potential candidate for solving a wide range of practical problems. It therefore deserves for pursuing and developing this approach for limit analysis of thin plate structures.
This paper further exploits the advantage of a rotation-free isogeometric approach to the assessment of collapse limit loads of plastic thin plates in bending. The kinematic formulation relies on the displacement (deflection) approximation using NURBS, resulting in both exact geometric representation and high-order approximations. Only deflection degrees of freedom are involved in the underlying optimization problem. This enables us to design the resulting optimization problem with a minimum size and to reduce computational cost. We adopt a simple procedure to eliminate rotational degrees of freedom on essential boundary conditions related to the constraint of normal slopes. The resulting non-smooth optimization problem is then written in the form of minimizing a sum of Euclidean norms so that it can be solved using highly efficient interior-point solvers. Several numerical examples are provided to show the reliability and accuracy of the present formulation.
The paper is arranged as follows: a brief review of B-spline and NURBS surfaces is described in the next section. This is followed by a section stating a rotation-free NURBS-based isogeometric formulation for limit analysis of thin plate problems. The solution procedure is given in the fourth section. Several numerical examples are illustrated in the fifth section. Finally, we close our paper with some concluding remarks.
2.1 A brief review of NURBS basis functions and surfaces
2.1.1 Knot vectors and basis functions and surfaces
2.1.2 NURBS surfaces
where P i are the control points, n denotes the number of control points and Ni,p(ξ) is the p th-degree B-spline basis function defined on the open knot vector.
where Ni,p(ξ) and Mj,q(η) are the B-spline basis functions defined on the knot vectors Ξ and , respectively.
where is the shape function associated with a node I. The superscript b indicates that is a B-spline shape function.
2.2 Rotation-free isogeometric formulation for upper bound limit analysis of plates
2.2.1 A background on limit analysis theorems of thin plates
Let be the mid-plane of a plate and be the transversal displacement velocity (or deflection velocity) in the z direction. Further, let us consider a kinematical boundary and a static boundary , where the subscript n stands for the outward normal vector. The general relations for thin Kirchhoff plates are described as follows.
where is the transverse load, λ is the collapse load multiplier and the differential operator ∇2 is defined by .
Flow rule and yield condition
The dissipation rate
Details on the derivation of the dissipation for plate problems can be found in . Let it be pointed out that, here, the velocity fields are supposed to be C1-continuous. In fact, more general fields presenting discontinuities of the normal rotation are possible. In this case, the expression of the dissipation power includes a supplementary term. For more details on this aspect, we refer to .
Let denote a space of kinematically admissible velocity field:
and Σ be an appropriate space of symmetric stress tensors and B is a set of essential boundary conditions defined in subsection ‘Essential boundary conditions’. More details on the mathematical formulations for limit analysis can be found in . The external work rate of a transversal force associated with a virtual plastic flow is expressed in the linear form as
The equilibrium equation is then described in the form of virtual work rate as follows:
Furthermore, the stresses m must satisfy the yield condition for the assumed material. This stress field belongs to a convex set, , obtained from the used field condition. For the von Mises criterion, one writes
If defining , the exact collapse multiplier λexact can be determined by solving any of the following optimization problems :
Problems (20) and (23) are known as static and kinematic principles of limit analysis, respectively. The limit load of both approaches converges to the exact solution. Herein, a saddle point () exists such that both the maximum of all lower bounds λ− and the minimum of all upper bounds λ+ coincide and are equal to the exact value λexact. In this work, we only focus on the kinematic formulation. Hence, problem (23) will be used to evaluate an upper-bound limit load factor using a NURBS-based isogeometric approach.
2.2.2 NURBS-based approximate formulation
Using the same NURBS basis functions, both the description of the geometry (or the physical point) and the velocity of the displacement field are expressed as
where n×m represent the number of basis functions, x T =(x,y) is the physical coordinates vector, N I (ξ,η) is the NURBS basis function and is the nodal value of at the control point I, respectively.
where N G=n e l×n G is the total number of Gauss points of the problem, nG is the number of Gauss points in each element, is the weight value at the Gauss point i and |J i | is the determinant of the Jacobian matrix computed at the Gauss point i.
where is the global basic function vector and () is the Gaussian quadrature point in a bi-unit parent element.
Since integral Equation 27 is not calculated exactly, it cannot be said that the formulation yields a strict upper bound using this formula. Nevertheless, the optimal velocity field is still kinematically admissible and corresponding compatible strain are obtained using the B i matrix so that a strict upper bound can be obtained provided that the dissipation is computed exactly a posteriori. However, in practice, there is practically no difference and the computed values can be considered as upper bounds.
2.2.3 Essential boundary conditions
In this part, we show how to impose essential boundary conditions of the isogeometric approach. For the sake of simplicity, we consider the following several Dirichlet boundary conditions (BCs):
where is the normal rotation constraint and x D are control points that define the essential boundary.
where the matrix Bec and vector bec of Equation 34 are given by
where row 1, row 2 to d+1 and row d+2 to 2d+1 in Bec matrix stand for the number of constraints related to an external work rate, the boundary control points and control points adjacent to the boundary, respectively, and is described as
where d denotes the number of control points defining the Dirichlet boundary with respect to a set of boundary control points .
It be can observed that the enforcement of essential boundary conditions using the rotation-free approach is simple and efficient in comparison with other numerical methods. For instance, readers can find more details on the advantages of this procedure in ,,,. In addition, IGA based on the Lagrange multiplier, penalty and collocation methods (see ) can be also used to enforce essential boundary conditions for the thin plate.
2.3 Solution procedure of the discrete problem
2.3.1 Second-order cone programming
where or are optimization variables, and the coefficients are , , , , and . For optimization problems in 2D or 3D Euclidean space, mdim=2 or mdim=3, respectively. When mdim=1, the SOCP problem reduces to a linear programming problem.
2.3.2 Solution procedure using second-order cone programming
The limit analysis problem, Equation 31, is a non-linear optimization problem with equality constraints. As stated before, the problem can be reduced to the problem of minimizing a sum of norms following the procedure described by Andersen et al. .
where the first constraint in Equation 44 represents quadratic cones. The total number of variables of the optimization problem is Nvar=NoDofs+4×NG where NoDofs is the total number of the degrees of freedom (DOFs) of the underlying problem. As a result, the optimization problem defined by Equation 44 can be effectively solved by the academic version of the Mosek optimization package .
3Results and discussion
In this section, we examine the performance of the present approach through the limit analysis of beams and plates. The computations are performed on a desktop computer with ADM Phenom II X6 (2.8GHz CPU, 16G RAM). For purpose of comparison with other published methods in the literature, we will restrict our interest to simply supported and clamped plates, which are the most frequently found case in practice. As shown in , the standard Gaussian quadrature rule (or nG=(p+1)×(p+1) Gauss points) is used to evaluate the integrals of NURBS elements of p degree. Just a precision, this is valid only for evaluating stiffness matrices and load vector for elastoplastic analyses, but here the dissipation involving a square root cannot be evaluated exactly using Gauss rules. However, we also can utilize this quadrature rule for limit analysis problem without much loss of accuracy of solution. It also is worth mentioning that the computational cost increases significantly when higher-order elements are used. This was pointed out that using the Gauss quadrature rule for NURBS elements is far from optimal. Hence, a simple and efficient quadrature algorithm , for NURBS-based isogeometric analysis will be recommended for our future research. For the limit problem of thin plates, we in this study employ only nG=p×p Gauss pointsa to compute the integral in Equation 27. We also exploit the so-called k-refinement approach, which is a unique characteristic of IGA as a flexible way for refinement and degree elevation for limit analysis problems. Note that with the same number of elements, the total number of DOFs of IGA is less than that of FEM. For all the examples, the von Mises criterion and perfectly rigid plastic material are used.
A comparison of the limit load factor of beams with m p = σ 0 b t 2 /4
Present (p = 2)
8.0007 (66 Dofs)
11.703 (514 Dofs)
16.063 (258 Dofs)
Present (p = 3)
8.0004 (67 Dofs)
11.687 (515 Dofs)
16.042 (259 Dofs)
3.2 Rectangular plates
The convergence of the limit load factor ( ) for a clamped square plate
Hodge and Belytschko 
The convergence of the limit load factor for a simply supported square plate
Hodge and Belytschko 
A comparison of the limit load factor for a square plate
Hodge and Belytschko 
Quadratic field (nonconforming)
Capsoni and Corradi 
Le et al. 
Bleyer and de Buhan 
Bleyer and de Buhan 
Zhou et al. 
Le et al. 
Le et al. 
Le et al. 
Enhanced Morley (EM)
Present (p = 2)
Present (p = 3)
The limit load factor for a rectangular plate with a / b =2 and various condition boundaries
3.3 Rhombic plate
Results of the limit load factor for the rhombic plate
Capsoni and Silva []
Zhou et al. []
Present (p =2)
Present (p =3)
3.4 L-shaped plate
The limit load factor for an L-shaped plate
3.5 Circular plate
3.5.1 Circular plate subjected to uniform transverse loading
3.5.2 Circular plate subjected to non-uniform transverse loading
where a1, a2 and a3 are predefined constants. In the numerical calculation, the constants a1, a2 and a3 are chosen in such a way that a1 is fixed at value of 3 (a1=3) and a2,a3 vary. The geometry and material parameters are as given in the previous case. For illustration, we use a slightly fine mesh of 17×17 NURBS elements for quadratic and cubic elements.
The limit load factor λ cr / m p for a clamped circular plate subjected to a linear load
The limit load factor λ cr / m p for a clamped circular plate subjected to a parabolic load
We have for the first time presented a rotation-free isogeometric finite element approach for upper bound limit analysis of thin plate structures. The method was derived from the kinematic theorem and isogeometric finite elements. The underlying optimization formulation of limit analysis was transformed into the form of a second-order cone programming, and it was then solved by highly efficient interior-point solvers. The performance of the method is validated through benchmark problems of plastic thin plates. Through the examples tested, some concluding remarks can be given as follows:
Only deflection degrees of freedom were needed in the optimization problem. Thus, the method requires less variables than N4 element (or C0-FEM), HCT element (or C1-FEM) and C1 natural element (C1-NEM). As a result, it is promising to provide an effective way to solve large-scale plate problems.
The essential boundary conditions are easily imposed in the context of the rotation-free isogeometric approach.
Beyond h-and p-refinement schemes currently available in the traditional FEM, the present approach was found to be more efficient with k-refinement type for limit analysis of thin plates.
Numerical results showed that the present method provides upper bound estimates of collapse limit loads, that proves the stability of the method. Also, the proposed method exhibited very good agreement with several published results in the literature for different benchmark problems. It seems, in particular, very efficient for the L-shaped plate problem which presents a singularity near the corner.
In present work, only benchmark problems were used to show the performance of the proposed formulation. However, we believe that the methodology is generalizable for large-scale plate problems in practice. Although the present method achieved high reliability, its computational cost is still significant due to an excessive overhead of control points for very uniformly refined meshes. It would therefore be interesting to associate the present method with adaptive local refinement procedures ,. This is a work in progress and our findings will be devoted in a forthcoming paper.
a This helps to reduce the size of the optimization problem without loss of accuracy of solution as it will be shown later.
6.1 Knot vectors and control points for NURBS objects
6.1.1 Circular plate
Control points and weights for a disk of radius 0.5
6.1.2 L-shaped plate
Control points for the L-shaped plate
This research is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 107.02-2014.24. The support is gratefully acknowledged.
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