Design of pipeline opening layout through level set topology optimization

Abstract

Perforated pipeline structure is widely utilized in the oil industry for its special functionality of communicating media with the ambient environment. A typical application is the slotted liner in SAGD (Steam Assisted Gravity Drainage) process, where the pipeline structure is manufactured with open slots to spread hot steam and collect the melted oil. Generally, a dense opening layout is employed to reduce flow resistance. On the other hand, inclusion of the many openings severely reduces the structural strength and stiffness, which causes the pipeline prone to deformation or even failure. Therefore, there exist the two conflicting requirements for design of the pipeline opening layout, and an interesting solution is proposed in this paper. To be specific, the pipeline structure is discretized into shell elements which are categorized into multiple types: without opening, with opening type 1, with opening type 2, etc. These element types are treated as different material phases, and design of the pipeline opening layout is transformed into a multi-material topology optimization problem. Multi-material level set method is employed to solve it, subject to the compliance minimization objective. In addition, a lower bound of opening quantity is applied by properly configuring the material fraction constraint, which ensures the low flow resistance. The effectiveness of the proposed method is proven through a few numerical case studies.

Appendix

To prove Eq. (26) equal to zero, we follow the derivations presented in (Guo et al. 2014). The derivation will be based on the 2D planar problem for the sake of simplicity, and it can be easily extended to shell problem by adding the design freedoms.

Through some mathematical manipulations (Guo et al. 2014), the following relationships can be derived:

$$ \begin{array}{l}{\displaystyle \underset{D}{\int }{\boldsymbol{D}}_s{\boldsymbol{e}}_s\left(\tilde{\boldsymbol{u}}\right){\boldsymbol{e}}_s\left(\nabla \tilde{\boldsymbol{w}}\cdot \boldsymbol{V}\right)H\left(\varPhi \right)d\varOmega }={\displaystyle \underset{D}{\int}\left(\boldsymbol{\sigma} \left(\tilde{\boldsymbol{u}}\right)\cdot \boldsymbol{n}\right)\cdot \left(\nabla \tilde{\boldsymbol{w}}\cdot \boldsymbol{V}\right)\delta \left(\varPhi \right)d\varOmega }+{\displaystyle \underset{D}{\int}\tilde{\boldsymbol{p}}\cdot \left(\nabla \tilde{\boldsymbol{w}}\cdot \boldsymbol{V}\right)H\left(\varPhi \right)d\varOmega}\hfill \\ {}{\displaystyle \underset{D}{\int }{{\boldsymbol{D}}_s}^{slot}{\boldsymbol{e}}_s\left(\tilde{\boldsymbol{u}}\right){\boldsymbol{e}}_s\left(\nabla \tilde{\boldsymbol{w}}\cdot \boldsymbol{V}\right)\left[1-H\left(\varPhi \right)\right]d\varOmega}\hfill \\ {}={\displaystyle \underset{D}{\int}\left[\boldsymbol{\sigma} \left(\tilde{\boldsymbol{u}}\right)\cdot \left(-\boldsymbol{n}\right)\right]\cdot \left(\nabla \tilde{\boldsymbol{w}}\cdot \boldsymbol{V}\right)\delta \left(-\varPhi \right)d\varOmega }+{\displaystyle \underset{D}{\int}\tilde{\boldsymbol{p}}\cdot \left(\nabla \tilde{\boldsymbol{w}}\cdot \boldsymbol{V}\right)H\left(-\varPhi \right)d\varOmega}\hfill \end{array} $$

(A1)

in which, $\tilde{\boldsymbol{p}}$ is the planar body force, and σ(ũ) is the second order Cauchy stress tensor. Accordingly, Eq. (37) can be derived as:

$$ \begin{array}{l}-{\displaystyle \underset{D}{\int }{\boldsymbol{D}}_s{\boldsymbol{e}}_s\left(\tilde{\boldsymbol{u}}\right){\boldsymbol{e}}_s\left(\nabla \tilde{\boldsymbol{w}}\cdot \boldsymbol{V}\right)H\left(\varPhi \right)d\varOmega }-{\displaystyle \underset{D}{\int }{{\boldsymbol{D}}_s}^{slot}{\boldsymbol{e}}_s\left(\tilde{\boldsymbol{u}}\right){\boldsymbol{e}}_s\left(\nabla \tilde{\boldsymbol{w}}\cdot \boldsymbol{V}\right)\left[1-H\left(\varPhi \right)\right]d\varOmega}\hfill \\ {}=-{\displaystyle \underset{D}{\int}\left(\boldsymbol{\sigma} \left(\tilde{\boldsymbol{u}}\right)\cdot \boldsymbol{n}\right)\cdot \left(\nabla \tilde{\boldsymbol{w}}\cdot \boldsymbol{V}\right)\delta \left(\varPhi \right)d\varOmega }-{\displaystyle \underset{D}{\int}\left[\boldsymbol{\sigma} \left(\tilde{\boldsymbol{u}}\right)\cdot \left(-\boldsymbol{n}\right)\right]\cdot \left(\nabla \tilde{\boldsymbol{w}}\cdot \boldsymbol{V}\right)\delta \left(-\varPhi \right)d\varOmega}\hfill \\ {}-{\displaystyle \underset{D}{\int}\tilde{\boldsymbol{p}}\cdot \left(\nabla \tilde{\boldsymbol{w}}\cdot \boldsymbol{V}\right)d\varOmega}\hfill \end{array} $$

(A2)

In Eq. (37), the first two terms at the right side could cancel each other at the material/material interface, and only remain at the fixed domain boundary ∂D. Therefore, Eq. (A2) can be further simplified into:

$$ \begin{array}{l}-{\displaystyle \underset{D}{\int }{\boldsymbol{D}}_s{\boldsymbol{e}}_s\left(\tilde{\boldsymbol{u}}\right){\boldsymbol{e}}_s\left(\nabla \tilde{\boldsymbol{w}}\cdot \boldsymbol{V}\right)H\left(\varPhi \right)d\varOmega }-{\displaystyle \underset{D}{\int }{{\boldsymbol{D}}_s}^{slot}{\boldsymbol{e}}_s\left(\tilde{\boldsymbol{u}}\right){\boldsymbol{e}}_s\left(\nabla \tilde{\boldsymbol{w}}\cdot \boldsymbol{V}\right)\left[1-H\left(\varPhi \right)\right]d\varOmega}\hfill \\ {}=-{\displaystyle \underset{D}{\int}\tilde{\boldsymbol{\tau}}\cdot \left(\nabla \tilde{\boldsymbol{w}}\cdot \boldsymbol{V}\right)d\varOmega }-{\displaystyle \underset{D}{\int}\tilde{\boldsymbol{p}}\cdot \left(\nabla \tilde{\boldsymbol{w}}\cdot \boldsymbol{V}\right)d\varOmega =}-{\displaystyle \underset{D}{\int}\tilde{\boldsymbol{p}}\cdot \left(\nabla \tilde{\boldsymbol{w}}\cdot \boldsymbol{V}\right)d\varOmega}\hfill \end{array} $$

(A3)

in which, $\tilde{\boldsymbol{\tau}}$ is the planar boundary traction force and it is acted on the non-design boundary. According to Eq. (A1), Eq. (26) can be proven to hold.

The same process can be employed to prove the last four terms at the left side of Eq. (28) equal to zero, as demonstrated in Eq. (A2).

$$ \begin{array}{l}-{\displaystyle \underset{D}{\int }{\boldsymbol{D}}_s{\boldsymbol{e}}_s\left(\tilde{\boldsymbol{u}}\right){\boldsymbol{e}}_s\left(\nabla \tilde{\boldsymbol{u}}\cdot \boldsymbol{V}\right)H\left(\varPhi \right)d\varOmega }-{\displaystyle \underset{D}{\int }{{\boldsymbol{D}}_s}^{slot}{\boldsymbol{e}}_s\left(\tilde{\boldsymbol{u}}\right){\boldsymbol{e}}_s\left(\nabla \tilde{\boldsymbol{u}}\cdot \boldsymbol{V}\right)\left[1-H\left(\varPhi \right)\right]d\varOmega}\hfill \\ {}-{\displaystyle \underset{D}{\int }{\boldsymbol{D}}_s{\boldsymbol{e}}_s\left(\tilde{\boldsymbol{w}}\right){\boldsymbol{e}}_s\left(\nabla \tilde{\boldsymbol{u}}\cdot \boldsymbol{V}\right)H\left(\varPhi \right)d\varOmega }-{\displaystyle \underset{D}{\int }{{\boldsymbol{D}}_s}^{slot}{\boldsymbol{e}}_s\left(\tilde{\boldsymbol{w}}\right){\boldsymbol{e}}_s\left(\nabla \tilde{\boldsymbol{u}}\cdot \boldsymbol{V}\right)\left[1-H\left(\varPhi \right)\right]d\varOmega}\hfill \\ {}=-{\displaystyle \underset{D}{\int}\tilde{\boldsymbol{p}}\cdot \left(\nabla \tilde{\boldsymbol{u}}\cdot \boldsymbol{V}\right)d\varOmega }-{\displaystyle \underset{D}{\int}\tilde{\boldsymbol{p}}\left(\tilde{\boldsymbol{w}}\right)\cdot \left(\nabla \tilde{\boldsymbol{u}}\cdot \boldsymbol{V}\right)d\varOmega =0}\hfill \end{array} $$

(A4)

Equation (A2) holds according to Eq. (A1), $\tilde{\boldsymbol{w}}=-\tilde{\boldsymbol{u}}$, and $\tilde{\boldsymbol{p}}\left(\boldsymbol{w}\right)=-\tilde{\boldsymbol{p}}$, in which $\tilde{\boldsymbol{p}}\left(\tilde{\boldsymbol{w}}\right)$ is the planar body force caused by $\tilde{\boldsymbol{w}}$.