Convergence analysis of Laplacian-based gradient elasticity in an isogeometric framework

Kolo, Isa; Askes, Harm; Borst, René de

doi:10.1016/j.finel.2017.07.006

Cited by 12 publications

(6 citation statements)

References 51 publications

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“…12,13 The pioneering work on the development of numerical methods in SGET was undertaken by Oden et al in their paper in 1970, 14 where they adopted the finite element method (FEM). Today, a lot of advanced numerical approaches and methods have been proposed in SGET that can be divided into the mixed FEM with C 0 -continuous interpolation and its variants, [15][16][17][18][19][20][21][22][23][24][25][26][27] FEM with C 1 -continuous interpolation, [28][29][30][31] isogeometric analysis, [32][33][34][35][36][37] singular boundary element methods (BEM) 38,39 and meshless Galerkin-type methods. [40][41][42] Among recent works, we can also mention the penalty three-dimensional elements approach 43 and the nonlocal operator method.…”

Section: Introductionmentioning

confidence: 99%

“…46 For example, it was recently shown that some of the earlier developed numerical methods in SGET cannot be applied to the problems with boundary conditions. 34 In the present work, we consider the indirect Trefftz method (TM), which is well known in classical theories. 47,48 In this method, the solution should be represented as a series of functions (trial functions) satisfying the governing differential equations of the model.…”

Section: Introductionmentioning

confidence: 99%

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Trefftz collocation method for two‐dimensional strain gradient elasticity

Solyaev

Lurie

2020

Numerical Meth Engineering

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Indirect Trefftz method is proposed for solving two‐dimensional boundary value problems of the strain gradient elasticity theory (SGET). A system of trial functions satisfying the fourth‐order equilibrium equations of SGET are developed based on the generalized Papkovich‐Neuber potentials. The classical part of the displacement solution is represented through the T‐complete system of functions satisfying the Laplace equation. The gradient part of the solution is represented through the system of heuristic functions satisfying the Helmholtz equation. The least squares collocation method is used to enforce the boundary conditions. Numerical examples are presented for the square domain under non‐uniform tensile and bending loads. It is shown, that the advantage of the presented method is that it allows to directly control the accuracy of the fulfillment of all nonstandard boundary conditions, that are prescribed in SGET on the surfaces and edges of the body.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Trefftz collocation method for two‐dimensional strain gradient elasticity

Solyaev

Lurie

2020

Numerical Meth Engineering

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“…To estimate the error, we use results of a mesh with 2 6 ×2 6 elements as the reference solution. Since the error estimation is now element-based, we use a slightly modified relation to calculate the relative error in each element [5]:φ…”

Section: Thick Hollow Cylinder Subjected To External Pressurementioning

confidence: 99%

“…Furthermore, NURBS have a natural higher-order character. This has motivated their use in higher-order gradient models where higher-order continuity is needed [2,3,4,5]. In these models, a length scale is incorporated in order to capture size effects and/or maintain a mesh-objective solution after the onset of softening.…”

Section: Introductionmentioning

confidence: 99%

Strain-gradient elasticity and gradient-dependent plasticity with hierarchical refinement of NURBS

Kolo

Chen

Borst

2019

Finite Elements in Analysis and Design

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Higher-order strain-gradient models are relevant for engineering materials which exhibit size-dependent behaviour as observed from experiments. Typically, this class of models incorporate a length scale-related to micromechanical material properties-to capture size effects, remove stress singularities, or regularise an ill-posed boundary value problem resulting from localisation of deformation. The higher-order continuity requirement on shape functions can be met using NURBS discretisation, as is considered herein. However, NURBS have a tensor-product nature which makes selective refinement cumbersome. To maintain accuracy and efficiency in analysis, a finer mesh may be required, to capture a localisation band, certain geometrical features, or in regions with high gradients. This work presents straingradient elasticity and strain-gradient plasticity, both of second-order, with hierarchically refined NURBS. Refinement is performed based on a multi-level mesh with element-wise hierarchical basis functions interacting through an inter-level subdivison operator. This ensures a standard finite-element data structure. Suitable marking strategies have been used to select elements for refinement. The capability of the numerical schemes is demonstrated with two-dimensional examples.

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“…A bottleneck for many gradient‐enhanced theories to gain wider acceptance is probably the C 1 requirement on the basis functions. As meshless and isogeometric methods can construct highly continuous basis functions, the integration of the weak form and the enforcement of boundary conditions in these method are not as straightforward as the finite element method. Furthermore, the support of the highly continuous basis functions is larger than those of the finite element model, leading to less sparse system matrices and long computing time.…”

Section: Introductionmentioning

confidence: 99%

Twenty‐four–DOF four‐node quadrilateral elements for gradient elasticity

Sze

2019

Numerical Meth Engineering

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Computational analysis of gradient elasticity often requires the trial solution to be C 1 , yet constructing simple C 1 finite elements is not trivial. In this paper, three four-node 24-DOF quadrilateral elements for gradient elasticity analysis are devised by generalizing some of the advanced element formulations for thin-plate analysis. These include the discrete Kirchhoff method, a relaxed hybrid-stress method, and the hybrid-stress method with equilibrating internal force modes. The first two methods start with the derivation of a C 0 displacement, which is quadratic complete in the Cartesian coordinates. In the first method, at the midside points are derived and interpolated together with those at the nodes. Strain is derived from the displacement interpolation yet the second-order displacement derivatives are derived from the displacement-gradient interpolation. In the second method, only the assumed constant double-stress modes are employed to enforce the continuity of the normal derivative of the displacement. In the third method, the equilibrating internal force modes require the C 1 displacement to be defined only along the element boundary and the domain interpolation can be avoided. Patch test involving linear stress and constant double stress as well as other tests are presented to validate the proposed elements. KEYWORDSC 1 displacement, discrete Kirchhoff, finite element, gradient elasticity, hybrid formulation INTRODUCTIONConventional elasticity theories employ strain as the only deformation measure and do not take the internal length scale of the material into account. They work well while the characteristic length of the deformation is considerably larger than the internal length scale. The requirement is met in conventional engineering applications of compact materials but not necessarily in, eg, natural materials, artificial cellular/foam materials, and micro/nano-devices. 1-6 A bottleneck for many gradient-enhanced theories to gain wider acceptance is probably the C 1 requirement on the basis functions. As meshless and isogeometric methods can construct highly continuous basis functions, 2,7-11 the integration of the weak form and the enforcement of boundary conditions in these method are not as straightforward as the finite element method. Furthermore, the support of the highly continuous basis functions is larger than those of the finite element model, leading to less sparse system matrices and long computing time. To date, the finite element method remains to be a well-accepted tool for engineering analyses. Unfortunately, the existing C 1 finite elements pose strict requirement on the mesh topology, 128 possess second-/third-order derivatives of the interpolated variable as the nodal degrees of freedom (DOFs), and/or employ heterosis nodes which are different from other element nodes in the nodal DOFs. 12 For instance, Agyris's three-node and six-node triangles use (u, u, x , u, y , u, xx , u, xy , u, yy ) as the parameters at the corner nodes and u, n , the displacement gradient along th...

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Convergence analysis of Laplacian-based gradient elasticity in an isogeometric framework

Cited by 12 publications

References 51 publications

Trefftz collocation method for two‐dimensional strain gradient elasticity

Trefftz collocation method for two‐dimensional strain gradient elasticity

Strain-gradient elasticity and gradient-dependent plasticity with hierarchical refinement of NURBS

Twenty‐four–DOF four‐node quadrilateral elements for gradient elasticity

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