Variational differential quadrature: A technique to simplify numerical analysis of structures

Shojaei, M. Faghih; Ansari, R.

doi:10.1016/j.apm.2017.02.052

Cited by 95 publications

(17 citation statements)

References 30 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The straightforward derivation process of high‐order tensors in equivalent matrix or vector quantities can be advantageous in the coding process of a numerical method. The discretization process is also performed using the derivative and integral operators of the VDQ method . The definition of these operators can be found in the Appendix.…”

Section: Matrix‐vector and Discretized Forms Of Formulationmentioning

confidence: 99%

“…In Equation ,

\overset{D}{true}_{X_{1}}

and

{\overset{D}{true}}_{X_{2}}

are the first‐order derivative operators of the VDQ technique with respect to X 1 and X 2 for a 2D domain (see the Appendix).…”

Section: Matrix‐vector and Discretized Forms Of Formulationmentioning

confidence: 99%

“…This makes the problem nonlinear. Also, in Equation ,

{\overset{2 d}{bold-scriptS}}_{X_{1} X_{2}; B_{0}}

denotes the integral operator of the VDQ method with respect to X 1 and X 2 in the 2D material configuration domain . Besides,

\begin{matrix} true \\ δ Π_{2} & = \int_{B_{0}} δ P : (\underline{boldH} - H) d A_{0} = \int_{B_{0}} δ \overset{true^}{boldP} \cdot (\underline{\overset{H}{true}} - \overset{true^}{boldH}) d A_{0} = \int_{B_{0}} δ {\overset{P}{true}}^{normalT} (\underline{\overset{H}{true}} - \overset{true^}{boldH}) d A_{0} \\ = δ {\overset{ℙ}{true}}^{normalT} ({double-struckK}_{boldPU} U + {double-struckK}_{boldPH} \overset{true^}{ℍ}), \end{matrix}

K_{PU} = ((), I_{4} ⊛ (⟨⟩, {\overset{S}{false}}_{X_{1} X_{2}; {frakturB}_{0}})) 𝔼_{H},

K_{PH} = - ((), I_{4} ⊛ (⟨⟩, {\overset{S}{false}}_{X_{1} X_{2}; {frakturB}_{0}})),

and …”

Section: Matrix‐vector and Discretized Forms Of Formulationmentioning

confidence: 99%

“…The presented approach is multifield because the displacement, the displacement gradient, and the first Piola‐Kirchhoff stress tensor are taken as independent unknowns according to the Hu‐Washizu principle. The numerical solution approach is developed based upon the variational differential quadrature (VDQ) method and a transformation technique. The main features of the present VDQ‐based approach are as follows: The tensor form of equations is replaced by a novel matrix‐vector format.…”

Section: Introductionmentioning

confidence: 99%

“…Hence, VDQ is locking free. It should be noted that locking might stem from improper choice of shape functions from ansatz FE spaces for the fields describing the kinematics and kinetics of deformations. In contrast to FEM, the VDQ method has not the assemblage process, making it much easier to implement. In comparison with the mixed finite element formulations, it does not involve the complexities related to considering several degrees of freedom for each element in the finite element exterior calculus. It is able to accurately solve the benchmark problems in which other standard methods encounter unphysical instabilities. In addition, the benefits of the VDQ method in terms of high degree of accuracy and fast convergence as compared to FEM have been fully discussed in the work of Faghih Shojaei and Ansari …”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

A VDQ‐based multifield approach to the 2D compressible nonlinear elasticity

Hassani

Ansari

Rouhi

2019

Numerical Meth Engineering

Self Cite

View full text Add to dashboard Cite

Summary A numerical multifield methodology is developed to address the large deformation problems of hyperelastic solids based on the 2D nonlinear elasticity in the compressible and nearly incompressible regimes. The governing equations are derived using the Hu‐Washizu principle, considering displacement, displacement gradient, and the first Piola‐Kirchhoff stress tensor as independent unknowns. In the formulation, the tensor form of equations is replaced by a novel matrix‐vector format for computational purposes. In the solution strategy, based on the variational differential quadrature (VDQ) technique and a transformation procedure, a new numerical approach is proposed by which the discretized governing equations are directly obtained through introducing derivative and integral matrix operators. The present method can be regarded as a viable alternative to mixed finite element methods because it is locking free and does not involve complexities related to considering several DOFs for each element in the finite element exterior calculus. Simple implementation is another advantage of this VDQ‐based approach. Some well‐known examples are solved to demonstrate the reliability and effectiveness of the approach. The results reveal that it has good performance in the large deformation problems of hyperelastic solids in compressible and nearly incompressible regimes.

show abstract

Section: Matrix‐vector and Discretized Forms Of Formulationmentioning

confidence: 99%

“…In Equation ,

\overset{D}{true}_{X_{1}}

and

{\overset{D}{true}}_{X_{2}}

are the first‐order derivative operators of the VDQ technique with respect to X 1 and X 2 for a 2D domain (see the Appendix).…”

Section: Matrix‐vector and Discretized Forms Of Formulationmentioning

confidence: 99%

“…This makes the problem nonlinear. Also, in Equation ,

{\overset{2 d}{bold-scriptS}}_{X_{1} X_{2}; B_{0}}

denotes the integral operator of the VDQ method with respect to X 1 and X 2 in the 2D material configuration domain . Besides,

\begin{matrix} true \\ δ Π_{2} & = \int_{B_{0}} δ P : (\underline{boldH} - H) d A_{0} = \int_{B_{0}} δ \overset{true^}{boldP} \cdot (\underline{\overset{H}{true}} - \overset{true^}{boldH}) d A_{0} = \int_{B_{0}} δ {\overset{P}{true}}^{normalT} (\underline{\overset{H}{true}} - \overset{true^}{boldH}) d A_{0} \\ = δ {\overset{ℙ}{true}}^{normalT} ({double-struckK}_{boldPU} U + {double-struckK}_{boldPH} \overset{true^}{ℍ}), \end{matrix}

K_{PU} = ((), I_{4} ⊛ (⟨⟩, {\overset{S}{false}}_{X_{1} X_{2}; {frakturB}_{0}})) 𝔼_{H},

K_{PH} = - ((), I_{4} ⊛ (⟨⟩, {\overset{S}{false}}_{X_{1} X_{2}; {frakturB}_{0}})),

and …”

Section: Matrix‐vector and Discretized Forms Of Formulationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

A VDQ‐based multifield approach to the 2D compressible nonlinear elasticity

Hassani

Ansari

Rouhi

2019

Numerical Meth Engineering

Self Cite

View full text Add to dashboard Cite

show abstract

A novel differential‐integral quadrature method for the solution of nonlinear integro‐differential equations

Mohamed

Abo-Hashem

2021

Math Methods in App Sciences

View full text Add to dashboard Cite

In this work, we introduce an integration matrix operator that is fully consistent with the differentiation matrix operator defined by the differential quadrature method (DQM). Using these operators, a generic differential-integral quadrature method "DIQM" is proposed to directly discretize an integro-differential equation as a system of algebraic equations. To extend the applicability of the proposed DIQM to solve nonlinear and/or variable coefficients integrodifferential equation, some matrix manipulations are introduced. A stability analysis for Volterra integro-partial-differential equations is presented and the exponential convergence of the proposed method is examined. Various types of integro-differential equations are solved including ordinary/partial, linear/ nonlinear, Volterra parabolic/hyperbolic integro-differential equations with different boundary and initial conditions. Numerical results demonstrate the exponential convergence and the applicability of DIQM.

show abstract

Multi‐patch variational differential quadrature method for shear‐deformable strain gradient plates

Torabi

Niiranen

Ansari

2022

Numerical Meth Engineering

View full text Add to dashboard Cite

The integration of generalized differential quadrature techniques and finite element (FE) methods has been developed during the past decade for engineering problems within classical continuum theories. Hence, the main objective of the present study is to propose a novel numerical strategy called the multi-patch variational differential quadrature (VDQ) method to model the structural behavior of plate structures obeying the shear deformation plate theory within the strain gradient elasticity theory. The idea is to divide the two-dimensional solution domain of the plate model into sub-domains, called patches, and then to apply the VDQ method along with the FE mapping technique for each patch. The formulation is presented in a weak form and due to the C 1 -continuity requirements the corresponding compatibility conditions are applied through the patch interfaces. The Lagrange multiplier technique and the penalty method are implemented to apply the higher-order compatibility conditions and boundary conditions, respectively. To show the efficiency of the proposed method, numerical results are provided for plate structures with both regular and irregular solution domains. The provided numerical examples demonstrate the applicability and accuracy of the method in predicting the bending and vibration behavior of plate structures following the higher-order plate model.

show abstract

Variational differential quadrature: A technique to simplify numerical analysis of structures

Cited by 95 publications

References 30 publications

A VDQ‐based multifield approach to the 2D compressible nonlinear elasticity

A VDQ‐based multifield approach to the 2D compressible nonlinear elasticity

A novel differential‐integral quadrature method for the solution of nonlinear integro‐differential equations

Multi‐patch variational differential quadrature method for shear‐deformable strain gradient plates

Contact Info

Product

Resources

About