Functionally Graded Materials (FGM) are advanced class of engineering composites constituting of ... more Functionally Graded Materials (FGM) are advanced class of engineering composites constituting of two or more distinct phase materials described by continuous and smooth varying composition of material properties in the required direction. In this work, the eect of material homogenization scheme on exural response of a thin to moderately thick FGM plate has been studied. The plate is subjected to dierent loading and boundary conditions. The formulation is developed based on the First order Shear Deformation Theory (FSDT). The mechanical properties are assumed to vary continuously through the thickness of the plate and obey a power law distribution of the volume fraction of the constituents. The variation of volume fraction through the thickness is computed using two dierent homogenization techniques; namely rule of mixtures and MoriTanaka scheme. Comparative studies have been carried out to demonstrate the eciency of the present formulation. The results obtained from the two techniques have been compared with the analytical solutions available in literature. In addition to the above a parametric study bringing out the eect of boundary conditions, loads and power law index have also been presented.
In this paper, natural neighbor Galerkin meshless method is employed for adaptive analysis of pla... more In this paper, natural neighbor Galerkin meshless method is employed for adaptive analysis of plates and laminates. The displacement field and strain field of plate are based on Reissner–Mindlin plate theory. The interpolation functions employed here were developed by Sibson and based on natural neighbor coordinates. An adaptive refinement strategy based on recovery energy norm which is in turn based on natural neighbors is employed for analysis of plates. The present adaptive procedure is applied to classical plate problems subjected to in-plane loads. In addition to that the laminated composite plates with cutouts subjected to transverse loads are investigated. Influence of the location of the cutout and the boundary conditions of the plate on the results have been studied. The results obtained with present adaptive analysis are accurate at lower computational effort when compare to that of no adaptivity. Further, the adaptive analysis provided accurate magnitude of maximum stresses and their locations in the laminate plates with and without cutout subjected to transverse loads. Additionally, failure prone areas in the geometry of the plates subjected to loads are revealed with the adaptive analysis.
In this work we present an implicit nonlocal gradient damage model. A scalar damage variable whic... more In this work we present an implicit nonlocal gradient damage model. A scalar damage variable which is a function of monotonically increasing function of history parameter is introduced to account for damage. The nonlocal model accounts for spatial interaction of neighboring material element at different length scales. The history parameter accounting for damage is obtained by considering nonlocal equivalent strains. To study the influence of nonlocal nature of approximation method on the proposed damage model, for numerical implementation, in a Galerkin framework, the influence of C p−1 and C 0 continuous approximation schemes are explored. C p−1 approximations are achieved by using Bezier and B-spline basis functions. C 0 approximations are achieved by using standard Lagrangian basis functions. From numerical examples, it is observed that C p−1 approximations are more nonlocal and results in better results than C 0 approximations.
In this work, nonlocal nonlinear finite element analysis of laminated composite plates using Redd... more In this work, nonlocal nonlinear finite element analysis of laminated composite plates using Reddy's third-order shear deformation theory (TSDT) [1] and Eringen's nonlocality [2] is presented. The governing equations of third order shear deformation theory with the von Kármán strains are derived employing the Eringen's [2] stress-gradient constitu-tive model. The principle of virtual displacement is used to derive the weak forms, and the displacement finite element models are developed using the weak forms. Four-noded rectangular conforming element with 8 degrees of freedom per node has been used. The coefficients of stiffness matrix and tangent stiffness matrix are presented along with non-local force vector. The developed finite element model can be employed to capture the small scale deviations from local continuum models caused by material inhomogeneity and the inter atomic and inter molecular forces. Numerical examples are presented to illustrate the effects of nonlocality, anisotropy, and the von Kármán type nonlinearity on the bending behaviour of laminated composite plates.
Mechanics of Advanced Materials and Structures , 2017
1 Abstract In the present work, the close similarity that exists between Mindlin's strain gradien... more 1 Abstract In the present work, the close similarity that exists between Mindlin's strain gradient elasticity and Eringens nonlocal integro-differential model is explored. A relation between length scales of nonlocal-differential model and gradient elasticity model has been arrived. Further, a relation has also been arrived between the standard and non-standard boundary conditions in both the cases. C 0 based finite element methods are extensively used for the implementation of integro-differential equations. This results in standard diagonally dominant global stiffness matrix with off diagonal elements occupied largely by the kernel values evaluated at various locations. The global stiffness matrix is enriched in this process by nonzero off diagonal terms and helps in incorporation of the nonlocal effect, there by accounting the long range interactions. In this case the diagonally dominant stiffness matrix has a band width equal to influence domain of basis function. In such cases, a very fine discretization with larger number of degrees of freedom are required to predict nonlocal effect, thereby making it computationally expensive. In the numerical examples, both nonlocal-differential and gradient elasticity model are considered to predict the size effect of tensile bar example. The solutions to integro-differential equations obtained by using various higher order approximations are compared. Lagrangian, B` ezier and B-Spline approximations are considered for the analysis. It has been shown that such higher order approximations have higher inter-element continuity there by increasing the band width and the nonlocal character of the stiffness matrix. The effect of considering the higher-order and higher-continuous approximation on computational effort is made. In conclusion both the models predict size
In this paper, we present the non-local nonlinear finite element formulations for the case of non... more In this paper, we present the non-local nonlinear finite element formulations for the case of nonuni-form rotating laminated nano cantilever beams using the Timoshenko beam theory. The surface stress effects are also taken into consideration. Non-local stress resultants are obtained by employing Erin-gen's nonlocal differential model. Geometric nonlinearity is taken into account by using the Green Lagrange strain tensor. Numerical solutions of nonlinear bending and free vibration are presented. Parametric studies have been carried out to understand the effect of non-local parameter and surface stresses on bending and vibration behavior of cantilever beams. Also, the effects of angular velocity and hub radius on the vibration behavior of the cantilever beam are studied.
Boundary value problems for a softening material suffer from loss of uniqueness in the post-peak ... more Boundary value problems for a softening material suffer from loss of uniqueness in the post-peak regime. Numerical solutions to such problems shows mesh dependency due to lack of internal length scale in the formulation. A regularization method which introduces a characteristic length is required to get mesh independent results. A second gradient model introduces a characteristic length by taking into account the second gradient of the displacement in the principle of virtual work and thus regularizing the solution of the boundary value problem. In this work a second gradient nite element model has been developed. The regularization property of the method has been studied for elastoplastic and damage constitutive laws. It has been shown that mesh independent results can be achieved in this model. Even though unique solution is not achieved a nite number of solutions have been obtained from the proposed model.
In the present work, an r-h adaptive isogeometric analysis is proposed for plane elasticity probl... more In the present work, an r-h adaptive isogeometric analysis is proposed for plane elasticity problems. For performing the r-adaption, the control net is considered to be a network of springs with the individual spring stiffness values being proportional to the error estimated at the control points. While preserving the boundary control points, relocation of only the interior control points is made by adopting a successive relaxation approach to achieve the equilibrium of spring system. To suit the non-interpolatory nature of the isogeometric approximation , a new point-wise error estimate for the h-refinement is proposed. To evaluate the point-wise error, hierarchical B-spline functions in Sobolev spaces are considered. The proposed adaptive h-refinement strategy is based on using De-Casteljau's algorithm for obtaining the new control points. The subsequent control meshes are thus obtained by using a recursive subdivision of reference control mesh. Such a strategy ensures that the control points lie in the physical domain in subsequent refinements, thus making the physical mesh to exactly interpolate the control mesh and thereby allowing the exact imposition of essential boundary conditions in the classical isogeomet-ric analysis. The combined r-h adaptive refinement strategy results in better convergence characteristics with reduced errors than r-or 1 h-refinement. Several numerical examples are presented to illustrate the efficiency of the proposed approach.
We present a natural element method to treat
higher-order spatial derivatives in the Cahn–Hilliar... more We present a natural element method to treat higher-order spatial derivatives in the Cahn–Hilliard equation. The Cahn–Hilliard equation is a fourth-order nonlinear partial differential equation that allows to model phase separation in binary mixtures. Standard classical C0-continuous finite element solutions are not suitable because primal variational formulations of fourth-order operators are well-defined and integrable only if the finite element basis functions are piecewise smooth and globally C1-continuous. To ensure C1-continuity, we develop a natural-element-based spatial discretization scheme. The C1-continuous natural element shape functions are achieved by a transformation of the classical Farin interpolant, which is basically obtained by embedding Sibsons natural element coordinates in a Bernstein–Bézier surface representation of a cubic simplex. For the temporal discretization, we apply the (second-order accurate) trapezoidal time integration scheme supplemented with an adaptively adjustable time step size. Numerical examples are presented to demonstrate the efficiency of the computational algorithm in two dimensions. Both periodic Dirichlet and homogeneous Neumann boundary conditions are applied. Also constant and degenerate mobilities are considered.We demonstrate that the use of C1-continuous natural element shape functions enables the computation of topologically correct solutions on arbitrarily shaped domains
In the present work we present a meshless natural neighbor Galerkin method for the bending and vi... more In the present work we present a meshless natural neighbor Galerkin method for the bending and vibration analysis of plates and laminates. The method has distinct advantages of geometric flexibility of meshless method. The compact support and the connectivity between the nodes forming the compact support are performed dynamically at the run time using the natural neighbor concept. By this method the nodal connectivity is imposed through nodal sets with reduced size, reducing significantly the computational effort in construction of the shape functions. Smooth non-polynomial type interpolation functions are used for the approximation of inplane and out of plane primary variables. The use of nonpolynomial type interpolants has distinct advantage that the order of interpolation can be easily elevated through a degree elevation algorithm, thereby making them suitable also for higher order shear deformation theories. The evaluation of the integrals is made by use of Gaussian quadrature defined on background integration cells. The plate formulation is based on first order shear deformation plate theory. The application of natural neighbor Galerkin method formulation has been made for the bending and free vibration analysis of plates and laminates. Numerical examples are presented to demonstrate the efficacy of the present numerical method in calculating deflections, stresses and natural frequencies in comparison to the Finite element method, analytical methods and other meshless methods available in the literature.
In this work a sensitivity analysis of linear elastic cracked structures using two
scale Generali... more In this work a sensitivity analysis of linear elastic cracked structures using two scale Generalized Finite Element Method (GFEM) is presented. The method is based on computation of material derivatives, mutual potential energies and direct di erentiation. In a computational setting the discrete form of the mutual potential energy release rate is simple and easy to calculate, as it only requires the multipli- cation of the displacement vectors and sti ness sensitivity matrices. By judiciously choosing the velocity eld, the method only requires displacement response in a sub-domain close to the crack tip, thus making the method computationally e- cient. The method thus requires an exact computation of displacement response in a sub-domain close to the crack tip. To this end in this study we have used a two scale GFEM for sensitivity analysis. GFEM is based on the enrichment of the classical nite element approximation. These enrichment functions incorporate the discontinuity response in the domain. Three numerical examples which comprise mode -I and mixed mode deformations are presented to evaluate the accuracy of the fracture parameters calculated by the proposed method.
Functionally Graded Materials (FGM) are advanced class of engineering composites constituting of ... more Functionally Graded Materials (FGM) are advanced class of engineering composites constituting of two or more distinct phase materials described by continuous and smooth varying composition of material properties in the required direction. In this work, the eect of material homogenization scheme on exural response of a thin to moderately thick FGM plate has been studied. The plate is subjected to dierent loading and boundary conditions. The formulation is developed based on the First order Shear Deformation Theory (FSDT). The mechanical properties are assumed to vary continuously through the thickness of the plate and obey a power law distribution of the volume fraction of the constituents. The variation of volume fraction through the thickness is computed using two dierent homogenization techniques; namely rule of mixtures and MoriTanaka scheme. Comparative studies have been carried out to demonstrate the eciency of the present formulation. The results obtained from the two techniques have been compared with the analytical solutions available in literature. In addition to the above a parametric study bringing out the eect of boundary conditions, loads and power law index have also been presented.
In this paper, natural neighbor Galerkin meshless method is employed for adaptive analysis of pla... more In this paper, natural neighbor Galerkin meshless method is employed for adaptive analysis of plates and laminates. The displacement field and strain field of plate are based on Reissner–Mindlin plate theory. The interpolation functions employed here were developed by Sibson and based on natural neighbor coordinates. An adaptive refinement strategy based on recovery energy norm which is in turn based on natural neighbors is employed for analysis of plates. The present adaptive procedure is applied to classical plate problems subjected to in-plane loads. In addition to that the laminated composite plates with cutouts subjected to transverse loads are investigated. Influence of the location of the cutout and the boundary conditions of the plate on the results have been studied. The results obtained with present adaptive analysis are accurate at lower computational effort when compare to that of no adaptivity. Further, the adaptive analysis provided accurate magnitude of maximum stresses and their locations in the laminate plates with and without cutout subjected to transverse loads. Additionally, failure prone areas in the geometry of the plates subjected to loads are revealed with the adaptive analysis.
In this work we present an implicit nonlocal gradient damage model. A scalar damage variable whic... more In this work we present an implicit nonlocal gradient damage model. A scalar damage variable which is a function of monotonically increasing function of history parameter is introduced to account for damage. The nonlocal model accounts for spatial interaction of neighboring material element at different length scales. The history parameter accounting for damage is obtained by considering nonlocal equivalent strains. To study the influence of nonlocal nature of approximation method on the proposed damage model, for numerical implementation, in a Galerkin framework, the influence of C p−1 and C 0 continuous approximation schemes are explored. C p−1 approximations are achieved by using Bezier and B-spline basis functions. C 0 approximations are achieved by using standard Lagrangian basis functions. From numerical examples, it is observed that C p−1 approximations are more nonlocal and results in better results than C 0 approximations.
In this work, nonlocal nonlinear finite element analysis of laminated composite plates using Redd... more In this work, nonlocal nonlinear finite element analysis of laminated composite plates using Reddy's third-order shear deformation theory (TSDT) [1] and Eringen's nonlocality [2] is presented. The governing equations of third order shear deformation theory with the von Kármán strains are derived employing the Eringen's [2] stress-gradient constitu-tive model. The principle of virtual displacement is used to derive the weak forms, and the displacement finite element models are developed using the weak forms. Four-noded rectangular conforming element with 8 degrees of freedom per node has been used. The coefficients of stiffness matrix and tangent stiffness matrix are presented along with non-local force vector. The developed finite element model can be employed to capture the small scale deviations from local continuum models caused by material inhomogeneity and the inter atomic and inter molecular forces. Numerical examples are presented to illustrate the effects of nonlocality, anisotropy, and the von Kármán type nonlinearity on the bending behaviour of laminated composite plates.
Mechanics of Advanced Materials and Structures , 2017
1 Abstract In the present work, the close similarity that exists between Mindlin's strain gradien... more 1 Abstract In the present work, the close similarity that exists between Mindlin's strain gradient elasticity and Eringens nonlocal integro-differential model is explored. A relation between length scales of nonlocal-differential model and gradient elasticity model has been arrived. Further, a relation has also been arrived between the standard and non-standard boundary conditions in both the cases. C 0 based finite element methods are extensively used for the implementation of integro-differential equations. This results in standard diagonally dominant global stiffness matrix with off diagonal elements occupied largely by the kernel values evaluated at various locations. The global stiffness matrix is enriched in this process by nonzero off diagonal terms and helps in incorporation of the nonlocal effect, there by accounting the long range interactions. In this case the diagonally dominant stiffness matrix has a band width equal to influence domain of basis function. In such cases, a very fine discretization with larger number of degrees of freedom are required to predict nonlocal effect, thereby making it computationally expensive. In the numerical examples, both nonlocal-differential and gradient elasticity model are considered to predict the size effect of tensile bar example. The solutions to integro-differential equations obtained by using various higher order approximations are compared. Lagrangian, B` ezier and B-Spline approximations are considered for the analysis. It has been shown that such higher order approximations have higher inter-element continuity there by increasing the band width and the nonlocal character of the stiffness matrix. The effect of considering the higher-order and higher-continuous approximation on computational effort is made. In conclusion both the models predict size
In this paper, we present the non-local nonlinear finite element formulations for the case of non... more In this paper, we present the non-local nonlinear finite element formulations for the case of nonuni-form rotating laminated nano cantilever beams using the Timoshenko beam theory. The surface stress effects are also taken into consideration. Non-local stress resultants are obtained by employing Erin-gen's nonlocal differential model. Geometric nonlinearity is taken into account by using the Green Lagrange strain tensor. Numerical solutions of nonlinear bending and free vibration are presented. Parametric studies have been carried out to understand the effect of non-local parameter and surface stresses on bending and vibration behavior of cantilever beams. Also, the effects of angular velocity and hub radius on the vibration behavior of the cantilever beam are studied.
Boundary value problems for a softening material suffer from loss of uniqueness in the post-peak ... more Boundary value problems for a softening material suffer from loss of uniqueness in the post-peak regime. Numerical solutions to such problems shows mesh dependency due to lack of internal length scale in the formulation. A regularization method which introduces a characteristic length is required to get mesh independent results. A second gradient model introduces a characteristic length by taking into account the second gradient of the displacement in the principle of virtual work and thus regularizing the solution of the boundary value problem. In this work a second gradient nite element model has been developed. The regularization property of the method has been studied for elastoplastic and damage constitutive laws. It has been shown that mesh independent results can be achieved in this model. Even though unique solution is not achieved a nite number of solutions have been obtained from the proposed model.
In the present work, an r-h adaptive isogeometric analysis is proposed for plane elasticity probl... more In the present work, an r-h adaptive isogeometric analysis is proposed for plane elasticity problems. For performing the r-adaption, the control net is considered to be a network of springs with the individual spring stiffness values being proportional to the error estimated at the control points. While preserving the boundary control points, relocation of only the interior control points is made by adopting a successive relaxation approach to achieve the equilibrium of spring system. To suit the non-interpolatory nature of the isogeometric approximation , a new point-wise error estimate for the h-refinement is proposed. To evaluate the point-wise error, hierarchical B-spline functions in Sobolev spaces are considered. The proposed adaptive h-refinement strategy is based on using De-Casteljau's algorithm for obtaining the new control points. The subsequent control meshes are thus obtained by using a recursive subdivision of reference control mesh. Such a strategy ensures that the control points lie in the physical domain in subsequent refinements, thus making the physical mesh to exactly interpolate the control mesh and thereby allowing the exact imposition of essential boundary conditions in the classical isogeomet-ric analysis. The combined r-h adaptive refinement strategy results in better convergence characteristics with reduced errors than r-or 1 h-refinement. Several numerical examples are presented to illustrate the efficiency of the proposed approach.
We present a natural element method to treat
higher-order spatial derivatives in the Cahn–Hilliar... more We present a natural element method to treat higher-order spatial derivatives in the Cahn–Hilliard equation. The Cahn–Hilliard equation is a fourth-order nonlinear partial differential equation that allows to model phase separation in binary mixtures. Standard classical C0-continuous finite element solutions are not suitable because primal variational formulations of fourth-order operators are well-defined and integrable only if the finite element basis functions are piecewise smooth and globally C1-continuous. To ensure C1-continuity, we develop a natural-element-based spatial discretization scheme. The C1-continuous natural element shape functions are achieved by a transformation of the classical Farin interpolant, which is basically obtained by embedding Sibsons natural element coordinates in a Bernstein–Bézier surface representation of a cubic simplex. For the temporal discretization, we apply the (second-order accurate) trapezoidal time integration scheme supplemented with an adaptively adjustable time step size. Numerical examples are presented to demonstrate the efficiency of the computational algorithm in two dimensions. Both periodic Dirichlet and homogeneous Neumann boundary conditions are applied. Also constant and degenerate mobilities are considered.We demonstrate that the use of C1-continuous natural element shape functions enables the computation of topologically correct solutions on arbitrarily shaped domains
In the present work we present a meshless natural neighbor Galerkin method for the bending and vi... more In the present work we present a meshless natural neighbor Galerkin method for the bending and vibration analysis of plates and laminates. The method has distinct advantages of geometric flexibility of meshless method. The compact support and the connectivity between the nodes forming the compact support are performed dynamically at the run time using the natural neighbor concept. By this method the nodal connectivity is imposed through nodal sets with reduced size, reducing significantly the computational effort in construction of the shape functions. Smooth non-polynomial type interpolation functions are used for the approximation of inplane and out of plane primary variables. The use of nonpolynomial type interpolants has distinct advantage that the order of interpolation can be easily elevated through a degree elevation algorithm, thereby making them suitable also for higher order shear deformation theories. The evaluation of the integrals is made by use of Gaussian quadrature defined on background integration cells. The plate formulation is based on first order shear deformation plate theory. The application of natural neighbor Galerkin method formulation has been made for the bending and free vibration analysis of plates and laminates. Numerical examples are presented to demonstrate the efficacy of the present numerical method in calculating deflections, stresses and natural frequencies in comparison to the Finite element method, analytical methods and other meshless methods available in the literature.
In this work a sensitivity analysis of linear elastic cracked structures using two
scale Generali... more In this work a sensitivity analysis of linear elastic cracked structures using two scale Generalized Finite Element Method (GFEM) is presented. The method is based on computation of material derivatives, mutual potential energies and direct di erentiation. In a computational setting the discrete form of the mutual potential energy release rate is simple and easy to calculate, as it only requires the multipli- cation of the displacement vectors and sti ness sensitivity matrices. By judiciously choosing the velocity eld, the method only requires displacement response in a sub-domain close to the crack tip, thus making the method computationally e- cient. The method thus requires an exact computation of displacement response in a sub-domain close to the crack tip. To this end in this study we have used a two scale GFEM for sensitivity analysis. GFEM is based on the enrichment of the classical nite element approximation. These enrichment functions incorporate the discontinuity response in the domain. Three numerical examples which comprise mode -I and mixed mode deformations are presented to evaluate the accuracy of the fracture parameters calculated by the proposed method.
The evolution of damage in laminated ber reinforced composites is a complex phenomena involving i... more The evolution of damage in laminated ber reinforced composites is a complex phenomena involving interaction of failure modes like matrix cracking, ber breakage, debonding and delamination. Damage initiation, its propagation and ultimate strength is of importance for developing a reliable and safer design and for utilizing them as a load bearing member. In the present work to understand and predict the mechanical behaviour of composite laminate of various congurations, a three dimensional nite element based progressive damage model is developed to delineate failure behaviour of a laminate in respective load-constraint conditions.The proposed model also helps to determine where and how failure occurs rst and how the damage evolves. Laminated composite plates of various stacking sequences have been analyzed. Hashin's and Puck's failure models are used for every case and their respective numerical results are compared. The inuences of the failure criteria and material degradation model are studied through
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Papers by Amirtham Rajagopal
higher-order spatial derivatives in the Cahn–Hilliard equation.
The Cahn–Hilliard equation is a fourth-order nonlinear
partial differential equation that allows to model phase separation
in binary mixtures. Standard classical C0-continuous
finite element solutions are not suitable because primal variational
formulations of fourth-order operators are well-defined
and integrable only if the finite element basis functions are
piecewise smooth and globally C1-continuous. To ensure
C1-continuity, we develop a natural-element-based spatial
discretization scheme. The C1-continuous natural element
shape functions are achieved by a transformation of the
classical Farin interpolant, which is basically obtained by
embedding Sibsons natural element coordinates in a
Bernstein–Bézier surface representation of a cubic simplex.
For the temporal discretization, we apply the (second-order
accurate) trapezoidal time integration scheme supplemented
with an adaptively adjustable time step size. Numerical
examples are presented to demonstrate the efficiency of the
computational algorithm in two dimensions. Both periodic
Dirichlet and homogeneous Neumann boundary conditions
are applied. Also constant and degenerate mobilities are considered.We demonstrate that the use of C1-continuous natural
element shape functions enables the computation of topologically
correct solutions on arbitrarily shaped domains
analysis of plates and laminates. The method has distinct advantages of geometric flexibility of
meshless method. The compact support and the connectivity between the nodes forming the compact
support are performed dynamically at the run time using the natural neighbor concept. By this method
the nodal connectivity is imposed through nodal sets with reduced size, reducing significantly the computational
effort in construction of the shape functions. Smooth non-polynomial type interpolation functions
are used for the approximation of inplane and out of plane primary variables. The use of nonpolynomial
type interpolants has distinct advantage that the order of interpolation can be easily elevated
through a degree elevation algorithm, thereby making them suitable also for higher order shear deformation
theories. The evaluation of the integrals is made by use of Gaussian quadrature defined on background
integration cells. The plate formulation is based on first order shear deformation plate theory.
The application of natural neighbor Galerkin method formulation has been made for the bending and free
vibration analysis of plates and laminates. Numerical examples are presented to demonstrate the efficacy
of the present numerical method in calculating deflections, stresses and natural frequencies in comparison
to the Finite element method, analytical methods and other meshless methods available in the
literature.
scale Generalized Finite Element Method (GFEM) is presented. The method is
based on computation of material derivatives, mutual potential energies and direct
di erentiation. In a computational setting the discrete form of the mutual potential
energy release rate is simple and easy to calculate, as it only requires the multipli-
cation of the displacement vectors and sti ness sensitivity matrices. By judiciously
choosing the velocity eld, the method only requires displacement response in a
sub-domain close to the crack tip, thus making the method computationally e-
cient. The method thus requires an exact computation of displacement response
in a sub-domain close to the crack tip. To this end in this study we have used a
two scale GFEM for sensitivity analysis. GFEM is based on the enrichment of the
classical nite element approximation. These enrichment functions incorporate the
discontinuity response in the domain. Three numerical examples which comprise
mode -I and mixed mode deformations are presented to evaluate the accuracy of
the fracture parameters calculated by the proposed method.
higher-order spatial derivatives in the Cahn–Hilliard equation.
The Cahn–Hilliard equation is a fourth-order nonlinear
partial differential equation that allows to model phase separation
in binary mixtures. Standard classical C0-continuous
finite element solutions are not suitable because primal variational
formulations of fourth-order operators are well-defined
and integrable only if the finite element basis functions are
piecewise smooth and globally C1-continuous. To ensure
C1-continuity, we develop a natural-element-based spatial
discretization scheme. The C1-continuous natural element
shape functions are achieved by a transformation of the
classical Farin interpolant, which is basically obtained by
embedding Sibsons natural element coordinates in a
Bernstein–Bézier surface representation of a cubic simplex.
For the temporal discretization, we apply the (second-order
accurate) trapezoidal time integration scheme supplemented
with an adaptively adjustable time step size. Numerical
examples are presented to demonstrate the efficiency of the
computational algorithm in two dimensions. Both periodic
Dirichlet and homogeneous Neumann boundary conditions
are applied. Also constant and degenerate mobilities are considered.We demonstrate that the use of C1-continuous natural
element shape functions enables the computation of topologically
correct solutions on arbitrarily shaped domains
analysis of plates and laminates. The method has distinct advantages of geometric flexibility of
meshless method. The compact support and the connectivity between the nodes forming the compact
support are performed dynamically at the run time using the natural neighbor concept. By this method
the nodal connectivity is imposed through nodal sets with reduced size, reducing significantly the computational
effort in construction of the shape functions. Smooth non-polynomial type interpolation functions
are used for the approximation of inplane and out of plane primary variables. The use of nonpolynomial
type interpolants has distinct advantage that the order of interpolation can be easily elevated
through a degree elevation algorithm, thereby making them suitable also for higher order shear deformation
theories. The evaluation of the integrals is made by use of Gaussian quadrature defined on background
integration cells. The plate formulation is based on first order shear deformation plate theory.
The application of natural neighbor Galerkin method formulation has been made for the bending and free
vibration analysis of plates and laminates. Numerical examples are presented to demonstrate the efficacy
of the present numerical method in calculating deflections, stresses and natural frequencies in comparison
to the Finite element method, analytical methods and other meshless methods available in the
literature.
scale Generalized Finite Element Method (GFEM) is presented. The method is
based on computation of material derivatives, mutual potential energies and direct
di erentiation. In a computational setting the discrete form of the mutual potential
energy release rate is simple and easy to calculate, as it only requires the multipli-
cation of the displacement vectors and sti ness sensitivity matrices. By judiciously
choosing the velocity eld, the method only requires displacement response in a
sub-domain close to the crack tip, thus making the method computationally e-
cient. The method thus requires an exact computation of displacement response
in a sub-domain close to the crack tip. To this end in this study we have used a
two scale GFEM for sensitivity analysis. GFEM is based on the enrichment of the
classical nite element approximation. These enrichment functions incorporate the
discontinuity response in the domain. Three numerical examples which comprise
mode -I and mixed mode deformations are presented to evaluate the accuracy of
the fracture parameters calculated by the proposed method.