Title 直交基底気泡関数要素安定化法による流体解析 ( 計算科学の基盤技術としての高速アルゴリズムとその周辺 ) Author(s) 松本 純一 Citation 数理解析研究所講究録 (2008) 1614: 187-198 Issue Date 2008-10 URL http://hdlhandlenet/2433/140100 Right Type Departmental Bulletin Paper Textversion publisher Kyoto University
10 12 $[$ 13 $[$ 1 1614 2008 187-198 187 Orthogonal Basis Bubble Function Element Stabilization Method for Fluid Analysis ) (Junichi Matsumoto) Advanced Manufacturing Research Institute National Institute of Advanced Industrial Science and Technology (AIST) Summary A two-level three-level formulation of finite element method with bubble function is proposed for the incompressible Navicr-Stokes equations Numerically spatial discretization is applied to the mixed interpolation for the velocity and pressure fields by bubble element and linear element respectively Numerical solutions for shape optimization with shape smoothing based on selective lumping method of flow past a circular cylinder are treated in this paper The purpose of the study is to formulate and solve an analysis of shape optimization for Navier-Stokes equations with unsteady flow To improve efficiency stability and accuracy of the calculation the mixed interpolation that uses an orthogonal basis bubble function element stabilization method for the state equations of incompressible viscous fluid is applied To validate the present method shape optimizatim for drag force of flow past a circular cylinder with periodic flow is analyzed Keywords: Navier-Stokes equations shape optimization shape smoothing bubble function element stabilization method orthogonal basis bubble function element 1 Navier-Stokes 3 $]$ $]$ $]$ $]$ Stokes $[$2 $[$3 $[$4 $[5][6 $ Navier-Stokes $]$ $[$8$]$ $[$9$]$ $[$ $]$ $]$ $[$ $]$ $[$7 $[$11 Navier-Stokes $]$ ( ) [13][14][15] Navier-Stokes [16] $[17 $
$\Gamma_{2}$ 188 2 $\dot{u}_{i}+u_{j}u_{ij}+p_{i}-\nu(u_{ij}+u_{ji})_{j}=0$ $in$ (1) $u_{ii}=0$ $in$ (2) $\Gamma$ $u_{i}$ $p$ $\nu=1/re$ $Re$ $\Gamma_{1}$ $u_{i}=\hat{u}_{i}$ $on$ $\Gamma_{1}$ (3) $\{-p\delta_{ij}+\nu(u_{ij}+u_{ji})\}\cdot n_{j}=t_{i}$ $on$ $\Gamma_{2}$ (4) $\delta_{ij}$ $\Gamma_{2}$ $n_{j}$ 3 31 MINI MINI [14] $]$ SUPG$/PSPG$ [18] BTD $[$19 MINI ( ) ( ) [15] 1: MINI MINI 1 ( 1 )
$\langle\dot{u}_{i}^{h}$ 189 32 $]$ 2 3 2 3 $[$15 $\overline{v}_{i}^{h}q^{h}$ 1 $V_{i}^{h }\hat{v}_{i}^{h }$ $\overline{v}_{i}^{h}=\{\overline{v}_{i}^{h}\in(h_{0}^{1}(\omega))^{2}\overline{v}_{i}^{h} _{\Omega_{\epsilon}}\in(P1(\Omega_{e}))^{2}\}$ (5) $V_{i}^{h^{l}}=\{v_{i}^{h }\in(h_{0}^{1}(\omega))^{2} v_{i}^{h } _{\zeta l_{e}}\in\phi_{b}v_{bi} v_{bi}^{j}\in R^{2}\}$ (6) $\hat{v}_{i}^{h }=\{\hat{v}_{i}^{h }\in(h_{0}^{1}(\omega))^{2}\hat{v}_{i}^{h } _{\Omega_{e}}\in\varphi Bv_{Bi}^{l} v_{\acute{b}i}\in R^{2}\}$ (7) $Q^{h}=\{q^{h}\in H_{0}^{1}(\Omega)$ $q^{h} _{\Omega_{e}}\in P1(\Omega_{e})$ $/r\iota^{q^{h}d\omega=0\}}$ (8) $\phi_{b}$ $\varphi B$ $\Omega_{e}$ 2 ( ) 3 ( ) (9)(10) $\langle\phi_{b}$ $1 \rangle_{\omega_{e}}=\vert\phi_{b}\vert_{\zeta)_{e}}^{2}=\frac{n+1}{n+2}\mathcal{a}_{e}$ (9) $\langle$ 1 (10) $\varphi B\rangle_{\Omega_{e}}=\langle\phi_{B\varphi B})_{\Omega_{e}}=0$ $V_{h} \hat{v}_{h} $ $\Omega_{e}NA_{e}$ $V_{i}^{h}=\overline{V}_{i}^{h}\oplus V_{i}^{h }$ $Q^{h}$ $(u_{i}^{h}p^{h})\in V_{i}^{h}xQ^{h}$ $\langle\dot{u}_{i}^{h}\hat{v}_{i}^{h})+\langle u_{j}^{h}u_{ij}^{h}\hat{v}_{i}^{h}\rangle+\langle p_{i}^{h}\hat{v}_{i}^{h}\rangle-\langle\nu(u_{ij}^{h}+u_{ji}^{h})_{j}\hat{v}_{i}^{h}\rangle=0$ $\forall\hat{v}_{i}^{h}\in\hat{v}_{i}^{h}$ (11) $\langle u_{ii}^{h}$ $q^{h})=0$ $\forall q^{h}\in Q^{h}$ (12) $\langle u$ $v N \rangle=\sum_{e=1}^{n_{e}}\langle u$ $v \rangle_{\zeta\}_{e}}=\sum_{e=1}^{n_{e}}l_{r\iota_{e}^{uv}}d\zeta)$ $u_{i}^{h}$ $V_{i}^{h}$ $\hat{v}_{i}^{h}=\overline{v}_{i}^{h}\oplus\{v_{i}^{h }+\hat{v}_{i}^{h };v_{i}^{h } _{\Omega}$ $+\hat{v}_{i}^{h^{l}} r\iota$ $=(\phi_{b}+\varphi B)v_{Bi} \}$ $\hat{v}_{i}^{h}$ $\overline{u}_{i}^{h}\overline{v}_{i}^{h}\in\overline{v}_{i}^{h}$ 1 $u_{i}^{h }$ $v_{i}^{h^{f}}\in V_{i}^{h }\hat{v}_{i}^{h }\in\hat{v}_{i}^{h }$ $u_{i}^{h}=\overline{u}_{i}^{h}+u_{i}^{h }\hat{v}_{i}^{h}=\overline{v}_{i}^{h}+v_{i}^{h }+\hat{v}_{i}^{h }=v_{i}^{h}+\hat{v}_{i}^{h }$ (13) $\langle\dot{u}_{i}^{h}$ $v_{i}^{h}\rangle+\langle u_{j}^{h}u_{ij}^{h}$ $v_{i}^{h})+\langle p_{i}^{h}$ $v_{i}^{h}\rangle-\langle\nu(u_{ij}^{h}+u_{ji}^{h})_{j}$ $v_{i}^{h}\rangle$ $+ \sum_{e=1}^{n_{e}}\langle\nu_{i}^{l}(u_{ij}^{h }+u_{ji}^{h })$ $v_{ij}^{h })_{\Omega_{e}}=0$ $\forall v_{i}^{h}\in V_{i}^{h}$ (14) $\langle u_{ii}^{h}$ $q^{h}\rangle=0$ $\forall q^{h}\in Q^{h}$ (15) $\nu_{\acute{i}};=\langle\dot{u}_{i}^{h}+u_{j}^{h}u_{ij}^{h}+p_{i}^{h}-\nu(u_{ij}^{h}+u_{ji}^{h})_{j}$ $\varphi B\rangle_{\Omega_{e}}/\langle(u_{ij}^{h }+u_{ji}^{h })$ $\phi_{bj}\rangle_{\omega_{e}}$ $\sum_{e=1}^{n_{e}}\langle\nu_{i} (u_{i}^{h_{j} }+u_{j_{i}}^{h })$ $v_{i}^{h_{j} }\rangle_{\omega_{e}}$ (14) 3 (14)(15) (16)(17) $v_{i}^{h}\rangle+\langle u_{j}^{h}u_{ij}^{h}$ $v_{i}^{h}\rangle-\langle p^{h}$ $v_{ii}^{h}\rangle+\langle\nu(\overline{u}_{ij}^{h}+\overline{u}_{ji}^{h})\overline{v}_{ij}^{h}\rangle$
$\overline{u}j$ 190 $+ \sum_{e=1}^{n_{e}}\langle(\nu+\nu_{i}^{l})(u_{ij}^{h }+u_{ji}^{h })$ $v_{ij}^{h })_{\Omega_{e}}=\langle t_{i}$ $v_{i}^{h}\rangle_{\gamma}$ $\forall v_{i}^{h}\in V_{i}^{h}$ (16) $\langle u_{ii}^{h}$ $q^{h}\rangle=0$ $\forall q^{h}\in Q^{h}$ (17) $\langle(\nu+\nu_{i}^{l})(u_{ij}^{h }+u_{ji}^{h })$ $v_{ij}^{h } \rangle r\iota_{e}=\frac{\langle\phi_{b}1\rangle_{\omega_{c}}^{2}}{a_{e}}\tau_{er}^{-1}\delta_{ij}u_{\acute{b}i}v_{bi} $ (18) $\tau_{er}=[(\frac{2 u_{i} }{h_{e}})^{2}+(\frac{4\nu}{h_{e}^{2}})^{2}]^{-1}2$ $h_{e}$ $A_{e}$ [14] Navicr-Stokes (16)(17) (16) 5 SUPG$/PSPG$ [18] BTD [19] 3 (18) [20][21][22] 4 41 (16)(17) $v_{i}^{t}(m\dot{u}_{i}+s(\overline{u}_{j})u_{i}-bp-m_{\gamma}t_{i})=0$ $in$ (19) $q^{t}(b^{t}u_{i})=0$ $in$ (20) $u_{i(t_{0})}=\hat{u}_{i0}$ $on$ (21) $M$ ( ) $S(\overline{u}j)$ 3 ) $B$ Mrti $J= \frac{1}{2}/t_{0}t_{f(e_{\gamma_{b}}^{t}m_{\gamma}t_{i}-d_{i})^{t}q_{i}(e_{\gamma_{b}}^{t}m_{\gamma}t_{i}-d_{i})dt}+\frac{1}{2}l_{t_{0}}^{t_{f}}(a_{c}-a_{0})q_{a}(a_{c}-a_{0})dt$ (22) $e_{\gamma_{b}}^{t}m_{\gamma}t_{i}=-/r_{b}t_{i}d\gamma$ $e_{\gamma_{b}}^{t}=[000-1 -1-1 \ldots 000]$ (23) $e_{\gamma}^{t}m_{\gamma}t_{i}$ $D_{i}$ $A_{C}$ $A_{0}$ $e_{\gamma_{b}}^{t}$ $B$ $-1$ $0$ $Q_{i}=1$ $Q_{a}$ (24) $Q_{a}=/t_{0}t_{f}(e_{\Gamma_{B}}^{T}M_{\Gamma}t_{i}-D_{i})^{T(0)}Q_{i}(e_{\Gamma_{B}}^{T}M_{\Gamma}t_{i}-D_{i})^{(0)}dt//t_{0}t_{f_{A_{C}^{2(0)}}}dt$ (24) (24) (0)
$ $ 191 42 $J$ $J^{*}=J+ \int_{t_{0}}^{t_{f}}\lambda_{u_{i}}^{t}(-s(\overline{u}_{j})u_{i}+bp+mrt_{i}-m\dot{u}_{i})dt+\int_{t_{0}}^{t_{f}}\lambda_{p}^{t}(b^{t}u_{i})dt$ (25) $\lambda_{ui}$ $\lambda_{p}$ $*$ $J$ $*$ * $\delta J^{*}=0$ (26) ( ) $M\dot{\lambda}_{u_{t}}-\tilde{S}(u_{j})^{T}\lambda_{u_{i}}+B\lambda_{p}=0$ $in$ (27) $B^{T}\lambda_{u_{i}}=0$ $in$ (28) $\lambda_{u_{i(t_{j})}}=0$ in (29) $\lambda_{ui}=-e_{\gamma_{b}}q_{i}(e_{\gamma_{b}}^{t}m_{\gamma}t_{i}-d_{i})$ $on$ $\Gamma$ (30) (27) $\overline{s}(u_{j})^{t}$ $S(\overline{u}_{j})u_{i}$ 43 431 (20) [10] Poisson $\tilde{u}_{i}^{n+1}$ $M \frac{\tilde{u}_{i}^{n+1}-u_{i}^{n}}{\delta t}+s(\overline{u}_{j}^{*})\tilde{u}_{i}^{n+1/2}-bp^{n}=mr_{2}t_{i}$ (31) $B^{T}M^{-1}B\Delta t(p^{n+1}-p^{n})=-b^{t}\tilde{u}_{i}^{n+1}$ (32) $M \frac{u_{i}^{n+1}-\tilde{u}_{i}^{n+1}}{\delta t}+\frac{1}{2}s(\overline{u}_{j}^{*})(u_{i}^{n+1}-\tilde{u}_{i}^{n+1})-b(p^{n+1}-p^{n})=0$ (33) $\overline{u}_{i}^{*}=\frac{1}{2}(3\overline{u}_{i}^{n}-\overline{u}_{i}^{n-1})\tilde{u}_{i}^{n+1/2}=\frac{1}{2}(\tilde{u}_{i}^{n+1}+u_{i}^{n})$ $n$ $\Delta t$ 432 (28)
$\tilde{d}_{j}^{(l)}$ $\text{ ^{}(l)}$ 192 Poisson $\tilde{\lambda}_{u_{i}}^{n-1}$ $M \frac{\tilde{\lambda}_{u_{i}}^{n-1}-\lambda u_{i^{n}}}{\delta t}+\tilde{s}(uj)^{t}\tilde{\lambda}_{u_{i}}^{n-1/2}-b\lambda p^{n}=0$ (34) $B^{T}M^{-1}B\Delta t(\lambda_{p}^{n-1}-\lambda_{p}^{n})=-b^{t}\tilde{\lambda}_{u_{i}}^{n-1}$ (35) $M \frac{\lambda_{u}^{n_{i}-1}-\tilde{\lambda}_{u_{i}}^{n-1}}{\delta t}+\frac{1}{2}\overline{s}(u_{j})^{t}(\lambda u_{i^{n-1}}-\tilde{\lambda}_{u_{i}}^{n-1})-b(\lambda_{p}^{n-1}-\lambda_{p}^{n})=0$ (36) $\tilde{\lambda}_{u_{i}}^{n-1/2}=\frac{1}{2}(\tilde{\lambda}_{u_{i}}^{n-1}+\lambda_{u_{t}}^{n})$ (32)(35) $M$ 44 $x_{j}^{(l+1)}=x_{j}^{(l)}+\alpha^{(l)}\tilde{d}_{j}^{(l)}$ (37) (38) d$)$ [17] $[- \{\frac{\partial 6^{\gamma}(\overline{u}_{j})}{(jx_{j}^{(l)}}1^{u_{i}}+\{\frac{\partial B}{\partial x_{j}^{(l)}}\}p]$ $d_{j}^{(l)}=-/_{t_{0}}t_{f}\{\lambda_{u}^{t}$ $+ \lambda_{p}^{t}\{\frac{\partial B^{T}}{\partial x_{j}^{(l)}}\}u_{i}+\{\frac{\partial A_{C}}{\partial x_{j}^{(l)}}\}q_{a}(a_{c}-a_{0})\}^{t(l)}dt$ (38) $\alpha^{(l)}$ $(l)$ Sakawa-Shindo [23] $\alpha^{(0)}$ $\alpha^{(0)}=\delta\hat{x}_{j\max}^{(1)}/\vert\tilde{d}_{j}^{(0)}\vert_{\infty}$ (39) $\Delta\hat{x}_{j_{\max}}^{(1)}$ $\Vert\cdot\Vert_{\infty}$ Ci $l=1$ $x_{j}^{(0)}$ 1 $l=0$ $u_{i}^{(l)}$ $p^{(l)}$ 2 (19)(20) 3 (22) $\lambda_{u_{i}}^{(l)}$ $\lambda_{p}^{(l)}$ 4 (27)(28) $x_{j}^{(l+1)}$ 5 (37) $e= x_{j}^{(l+1)}-x_{j}^{(l)} _{\infty}$ 6 $e<\epsilon$ $u_{i}^{(l+1)}$ 7 (19)(20) $p^{(l+1)}$ 8 (22) (l 1) $+$ $\alpha^{(l)}$ 9 $J^{(l+1)}\leq J^{(l)}$ $\alpha^{(l+1)}=\frac{10}{9}\alpha^{(l)}$ $l+1arrow l$ $\alpha^{(l)}=\frac{1}{2}\alpha^{(l)}$ 4 5
193 45 $d_{j}^{(l)}$ 1 Iterare: For $m=12\ldotsm_{s}$ do: $\overline{m}_{s}\tilde{d}_{j}^{(l)}=\tilde{m}_{s}d_{j}^{(l)}$ (40) $\tilde{d}_{j}^{(l)}arrow d_{j}^{(l)}$ (41) $\overline{m}_{s}\tilde{m}_{s}$ $m_{s}$ (2 3 ) $\tilde{m}_{s}=e_{8}\overline{m}_{\theta}+(1-e_{s})m_{s}$ $0\leq e_{\theta}\leq 1$ (42) $e_{s}$ ( ) $e_{s}=1$ $e_{s}=0$ $M_{s}$ (42) $\overline{m}_{s}\tilde{d}_{j}^{(l)}=\overline{m}_{s}d_{j}^{(l)}-(1-e_{s})(\overline{m}_{s}-m_{s})d_{j}^{(l)}$ (43) (43) $-(1-e_{s})(\overline{M}_{s}-M_{s})$ $l$ $l$ $M_{s}^{(e)}$ $\overline{m}_{s}^{(e)}$ $K_{s}^{(e)}$ $M_{s}^{(e)}= \frac{l}{6}\{\begin{array}{ll}2 11 2\end{array}\} \overline{m}_{s}^{(e)}=\frac{l}{6}\{3 3\} K_{s}^{(e)}= \frac{1}{l}[-11$ $-11$ (44) $M_{s}^{(e)}= \frac{\sqrt{3}}{48}l^{2}\{\begin{array}{lll}2 1 11 2 11 1 2\end{array}\}$ $j\overline{\nu f}_{s}^{(e)}=\frac{\sqrt{3}}{48}l^{2}\{\begin{array}{lll}4 4 4\end{array}\}$ $K_{s}^{(e)}= \frac{\sqrt{3}}{6}[-1-12$ $-1-12$ $-1-12$ (45) $(1-e_{s})(\overline{M}_{S}^{(e)}-M_{S}^{(e)})$ $(1-e_{s})( \overline{m}_{s}^{(e)}-m_{s}^{(e)})=(1-e_{s})\frac{l}{6}\{\begin{array}{ll}1-1-1 1\end{array}\}$ (46) $(1-e_{s})( \overline{m}_{s}^{(e)}-m_{s}^{(e)})=(1-e_{s})\frac{\sqrt{3}}{48}l^{2}[-1-12$ $-1-12$ $-1-12$ (47) (46) $(47)$ $(1-e_{s})(\overline{M}_{s}^{(e)}-$ (44) (45) $M(e))$ $e_{s}$ 1 $e_{\epsilon}$ (48) (49)(50) $0$ $\nu_{s}$ $e_{s}$ $(1-e_{e})(\overline{M}_{s}^{(e)}-M_{\delta}^{(e)})=\nu_{S}^{(e)}K_{s}^{(e)}$ (48)
194 $\nu_{s}^{(e)}=(1-e_{s})\frac{l^{2}}{6}=(1-e_{s})\frac{h_{e}^{2}}{6}$ $h_{e}=a_{\epsilon}$ $\mathcal{a}_{e}=l$ (49) $\nu_{\theta}^{(e)}=(1-e_{s})\frac{l^{2}}{8}=(1-e_{s})\frac{h_{e}^{2}}{6}\frac{\sqrt{3}}{2}$ $h_{e}=\sqrt{2a_{e}}$ $A_{e}= \frac{\sqrt{3}}{4}l^{2}$ (50) $h_{e}$ $e$ $A_{e}$ $A_{e}$ (49)(50) $e_{s}$ $e_{s}=1-\not\subset_{2}3$ $e_{\epsilon}=0$ (51) $e$ $\nu_{s}^{(\text{ })}=\frac{\sqrt{3}}{12}h_{e}^{2}$ (51) 52 ( ) 2(a)(b) 2(b) 1834 3500 56 $D$ 10 $A\backslash$ $Re=250$ ( ) 02 $t_{0}=200$ $t_{f}=300$ $t_{0}=200$ $\Delta\hat{x}_{jm}^{(1)}$ $10^{-2}10^{-5}$ $v=0$ $\epsilon$ $u=1$ $v=0$ $v=0$ (a) 51 (52)(53) $= \frac{1}{2}l_{t_{0}}^{t_{f}}(e_{\gamma_{b}}^{t}m_{\gamma}ti)^{t}qi(e_{\gamma_{b}}^{t}m_{\gamma}t_{1})dt+\frac{1}{2}/t_{0}t_{f}(a_{c}-a_{0})q_{a}(a_{c}-a_{0})dt$ (52)
Mrti) 195 (b) 2: $J= \frac{1}{2}/t_{0}t_{f}(e_{\gamma_{b}}^{t}m_{\gamma}t_{1}-d_{1})^{t}q_{1}(e_{\gamma_{b}}^{t}m_{\gamma}t_{1}-d_{1})dt+\frac{1}{2}/t_{0}t_{f_{a_{c}q_{a}a_{c}}}dt$ (53) $e_{\gamma_{b}}^{t}$ ( $A_{C}$ ( 2(b) ) (52) $D_{1}=0$ $A_{0}$ (53) $D_{1}$ (52) $A_{0}=0$ $A_{C}$ ( ) ( ) 52 45 $em_{s}$ 3 (52) $e=1m_{s}=1$ ( ) $e=1-c_{2}3$ $m_{s}$ 15101520 3 $D=10$ 3 (a) 3(b)(c) 3(d)(e)(f) 53 4 3(d) (52) (53) ( ) 4 (a) (b) ( ) 5 4(a)(b) ( ) ( ( ) (a)(b) $)$ -20% 300
196 (a) $e_{s}=1m_{s}=1$ 3: 4: $(e_{s}=1-\not\subset_{2}3 m_{s}=10)$ (b) ( ) (a) (b) ( ) 5: ( )
197 6 (52) $A_{C}$ $A_{0}$ 6(a) 4(a)(b) (a) (b) ( ) 6: $(e_{s}=1-l_{2^{3}} m_{s}=10)$ 6(b) ( ) -1% ( ) 6 Navier- Stokes $[17 $ [1] OPironneau:On optimum profiles in Stokes flow J Fluid Mech 59 Part 1 pp117 1973 [2] 0Pironneau:On optimum design in fluid mechanics J Fluid Mech 64 Part 1 pp97 1974 [3] JMBourot :On the numerical computation of the optimum profile in Stokcs flow J Fluid Mech 65 Part 3 pp513 1975 $]$ $[$4 :2 Stokes 193 pp115 1985 [5] : $A$ $60574$ pp 165 1994 [6] HOgawa and MKawahara:Shape optimization of Body Located in Incompressible Viscous Flow Based on Optimal Control Theory Int J Comput Fluid Dynamics 17 pp243-251 2003
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