Arabian Journal for Science and Engineering

, Volume 44, Issue 2, pp 1671–1684 | Cite as

Modeling and Optimization of Anisotropic Viscoelastic Porous Structures Using CQBEM and Moving Asymptotes Algorithm

  • Mohamed Abdelsabour FahmyEmail author
Research Article - Mechanical Engineering


The aim of this paper is to develop a novel numerical modeling optimization technique and implement it with the boundary element method (BEM) based on convolution quadrature method for studying optimization of anisotropic viscoelastic porous permeable structures. After applying the quadrature rule to the discretized boundary integral equation, a time stepping procedure based on the use of the linear multistep method is obtained. To obtain anisotropic fundamental solutions, the calculation of a double integral should be performed, but it increases the total computation time in the BEM. In order to overcome this problem and improve the computational efficiency of the formulation, our new proposed BEM technique is implemented. The method of implicit differentiation with respect to design variables has been implemented to calculate the displacements and pore pressure design sensitivities, and the effects of viscosity on these sensitivities are discussed. The resulting topology optimization problem has been solved using the method of moving asymptotes algorithm with adjoint variable method to optimize material distribution and find the influence of viscosity on the optimal design. The validity, efficiency and accuracy of the proposed BEM technique were confirmed by comparing obtained results for elliptical sandwich structure and multi-well reservoir with the corresponding results of finite difference method, lattice Boltzmann method and finite element method, which are special cases of our general and complex problem.


Convolution quadrature boundary element method Numerical modeling Topology optimization Anisotropic viscoelasticity Porous structure 

List of symbols


\(\phi \left( {1+{\bar{Q}} /{\bar{R}} } \right) \) Biot’s coefficient

\(C_{ajlg} \)

Constant elastic moduli (GPa)

\({\bar{C}} _{ajlg} \left( s \right) \)

Complex viscoelastic moduli in Laplace domain

\(C_{ajlg}^0 \)

Instantaneous stiffness tensor

\(C_{ajlg}^\infty \)

Equilibrium stiffness tensor

\({\mathbb {C}}\left( \tau \right) \)

Source term

\(c_{aj} \) and c

Free terms


\(\sqrt{\lambda _m /\rho }\) phase velocities

\(E_{jm} \)

Eigenvectors of \(\varGamma _{jl} \)

\(F_a \)

Bulk body forces \((\hbox {N})\)


Permeability \((\hbox {m}^{4}/\hbox {N s})\)


\(s/c_m \) wave numbers

\({\bar{k}} \)

Fixed value


Average length of edges for all the boundary elements


Number of design variables

\(N_e \)

Elements set whose distance from e is less than R


\(\left( {n_1,n_2,n_3 } \right) \) direction of wave propagation


Pore pressure (kPa)

\({\bar{Q}},{\bar{R}} \)

Solid–fluid coupling parameters


Minimum distance between considered boundary element and collocation point


Residual of the state equation

\(q_a \)

Fluid specific flux \((\hbox {m}^{3}/\hbox {s})\)

\(u_a \)

Solid displacement \((\hbox {m})\)

\(v_a \)

Fluid–solid displacements \((\hbox {m})\)

\(\upsilon _i\)

Element volume


\(V^{\mathrm{f}}+V^{\mathrm{s}}\) bulk volume \((\hbox {m}^{3})\)


Fluid volume \((\hbox {m}^{3})\)


Solid volume \((\hbox {m}^{3})\)

\(\fancyscript{w}\left( {\fancyscript{x}_i,~\fancyscript{x}_e } \right) \)

\(R-\Vert \fancyscript{x}_i -\fancyscript{x}_e \Vert \) weighting function

\(\fancyscript{x}_{{e}} \)

Corresponding coordinates of element e

\(\varGamma _{jl}\)

\({\bar{C}} _{ajlg} n_a n_g \) Christoffel tensor

\(\gamma \left( z \right) \)

Stability polynomial

\(\bar{\bar{\gamma }}\)

Solid volume fraction

\(\delta _{aj} \)

Kronecker delta function

\(\varepsilon _{kk} \)

Volumetric strain of the solid skeleton

\(\zeta \)

Fluid volume variation \((\hbox {m}^{3})\)

\(\lambda \)

Adjoint variable vector

\(\lambda _m \)

Eigenvalues of \(\varGamma _{jl} \)

\({\mu }\)

Shape factor

\(\xi _i^{\left( e \right) } \)

Coordinates of the corresponding corner nodes

\(\rho \)

\(\rho _\mathrm{s} \left( {1-\phi } \right) +\phi \rho _\mathrm{f} \) bulk density (\(\hbox {kg}/\hbox {m}^{3})\)

\(\rho _\mathrm{s}\)

\(\mu \phi \rho _{{\mathcal {F}}} \) solid density (\(\hbox {kg}/\hbox {m}^{3})\)

\(\rho _{{\mathcal {F}}} \)

Fluid density (\(\hbox {kg}/\hbox {m}^{3})\)

\({\bar{\rho }} _e \)

Standard filter densities

\(\sigma _{aj} \)

Total stress tensor (\(\hbox {N}/\hbox {m}^{2})\)

\(\tau \)

Time \(\left( \hbox {s} \right) \)

\(\phi \)

\(\frac{V^{\mathrm{f}}}{V}\) porosity

\(\varphi \)

Interpolation function

\(\varOmega \)

Region bounded by boundary \(\varGamma \)


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Copyright information

© King Fahd University of Petroleum & Minerals 2018

Authors and Affiliations

  1. 1.Jamoum University CollegeUmm Al-Qura UniversityJamoumm, MakkahSaudi Arabia
  2. 2.Faculty of Computers and InformaticsSuez Canal UniversityIsmailiaEgypt

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