STORX: MATLAB’da shakl va topologiya optimallashtirishini birlashtiruvchi ochiq manbali obyektga yo‘naltirilgan platform...

STORX — Shape and Topology Optimization for Research and Experimentation — parametrik shakl optimallashtirish, level-set shakl optimallashtirish va bir nechta topologiya optimallashtirish oilalarini bitta MATLAB asosidagi obyektga yo‘naltirilgan arxitektura ostida birlashtiruvchi ochiq manbali hisoblash dizayni frameworkidir. Ishning markaziy hissasi yagona yangi algoritm emas; geometriya, chekli elementlar tahlili, holat tenglamasi, maqsad va cheklov funksionallari, sezgirlik tahlili, dizayn yangilanishi hamda ishlab chiqarish cheklovlarini umumiy abstrakt interfeyslar orqali birlashtiradigan dasturiy arxitekturadir.

Framework parametrik optimallashtirish, Hamilton–Jacobi level-set usullari, SIMP/RAMP zichlik interpolatsiyasi, ESO, BESO va Pareto tracingni bir xil fizik va sonli sharoitlarda taqqoslashga imkon beradi. Bundan tashqari ko‘p yuk holatlari, o‘z og‘irligi, statsionar issiqlik o‘tkazuvchanligi, kuchlanishni minimallashtirish, lokal hajm ulushi va Brinkman penalizatsiyasiga asoslangan suyuqlik topologiya optimallashtirish kengaytmalari ko‘rsatilgan.

Natijalar STORX maxsus top88 kabi ixcham kodlardan har doim tezroq ekanini ko‘rsatmaydi. Aksincha, umumiy obyektga yo‘naltirilgan struktura qo‘shimcha hisoblash xarajatiga ega: o‘rta to‘r o‘lchamlarida iteratsiya vaqti taxminan 65–68% uzunroq, 320×160 da farq taxminan 60%. Taxminan 50 ming elementli masalada vizualizatsiyasiz bir iteratsiya qariyb 0,4 s. Asosiy ustunlik hisoblash tezligi emas, balki bir xil solver infratuzilmasida turli geometriya, fizika, maqsad, cheklov va optimallashtirish usullarini almashtirish imkonidir.

Muammo va arxitektura

Shakl optimallashtirishda dizayn o‘zgaruvchilari teshik radiusi, chegara joylashuvi yoki chamfer kabi geometrik parametrlar bo‘lishi mumkin. Topologiya optimallashtirishda esa element zichligi, level-set maydoni yoki topologik sezgirlik materialning qaerda saqlanishini va yangi teshik/aloqa shakllanishini boshqaradi. STORX bu usullardagi umumiy hisoblash zanjirini — geometriya, diskretlashtirish, FEA, sezgirlik, filtr va dizayn yangilanishini — bitta obyektga yo‘naltirilgan API orqali ko‘rsatadi.

Asosiy sinflar: brep2d, gridMesher, fea2d, simulation2d, functional, mfgConstraints, parameterOpt2d, density2d, levelset2d, evolutionary2d va pareto2d. Yangi maqsad uchun evaluate/gradient, yangi ishlab chiqarish cheklovi uchun filterDesign/filterSensitivity interfeyslari yetarli bo‘lishi ko‘zda tutilgan.

B-Rep va FEA

2D Boundary Representation koordinatalar va line/arc segmentlar orqali umumiy tekislik geometriyalarini tasvirlaydi. Ichki teshiklar construction segmentlar bilan topologik bog‘lanadi.

\[ R_{\mathrm{el}}(d)=K_{\mathrm{el}}d-f_{\mathrm{el}}=0. \tag{1} \]

Gripper tekshiruvida 4.000 ga yaqin element, \(E=2\) GPa, \(\nu=0.35\), 10 N yuk ishlatilgan. Triangular FEA: \(1.00\times10^{-6}\) m va 6,87 MPa; grid FEA: \(1.05\times10^{-6}\) m va 6,31 MPa.

Parametrik shakl optimallashtirish

\[ \begin{aligned} \min_{p=[a,b,c,r]^\top}\;&C(d;p) \tag{2a}\\ K_{\mathrm{el}}(p)d-f_{\mathrm{el}}&=0 \tag{2b}\\ A-A_{\max}&\le0 \tag{2c}\\ a_{\min}\le a&\le a_{\max} \tag{2d}\\ b_{\min}\le b&\le b_{\max} \tag{2e}\\ c_{\min}\le c&\le c_{\max} \tag{2f}\\ r_{\min}\le r&\le r_{\max}. \tag{2g} \end{aligned} \]

\[ C_i'=f^\top u_i' \tag{3} \]

\[ K^{(k)}u_i'=-K_i'u^{(k)} \tag{4} \]

\[ K_i'= \frac{K^{(k)}_{i,\mathrm{pert}}-K^{(k)}}{\Delta p_i}. \tag{5} \]

Yarim analitik FD natijasi: \(p_0=(0.20,0.15,1.20,0.10)\) dan \(p_{FD}=(0.363,0.224,1.500,0.076)\); 118 iteratsiya, 1.070 FEA; maydon 1,906 dan 1,7995 m²; compliance 3,91 dan 3,95 N·m; yakuniy maksimal egilish taxminan 0,04 mm.

GlobalSearch: \(p_{GS}=(0.310,0.226,1.248,0.141)\), A=1,79 m², C=4,022 N·m, 305 FEA. MultiStart: \(p_{MS}=(0.327,0.138,1.466,0.155)\), A=1,798, C=4,001, 2.545 FEA. Random Search: \(p_{RS}=(0.359,0.150,1.337,0.150)\), A=1,779, C=4,021, 22 FEA.

Level-set shakl optimallashtirish

\[ \psi(x) \begin{cases} <0,&x\in\Omega\\ =0,&x\in\partial\Omega\\ >0,&x\notin\Omega\cup\partial\Omega \end{cases} \tag{6} \]

\[ \begin{aligned} \min_\psi\;&\phi(\psi)\\ |\Omega(\psi)|-V^*&\le0,\\ R_{el}(d)&=0. \end{aligned} \tag{7} \]

\[ \frac{\partial\psi}{\partial t} +\operatorname{sign}(\psi_0)(|\nabla\psi|-1)=0 \tag{8} \]

\[ \operatorname{sign}(\psi)= \frac{\psi}{\sqrt{\psi^2+|\nabla\psi|^2\epsilon^2}}. \tag{9} \]

Zichlikka asoslangan TO

\[ \begin{aligned} \min_\rho\;&\phi(d;\rho) \tag{10a}\\ \sum_e\rho_ev_e-V^*&\le0 \tag{10b}\\ K_{el}(\rho)d-f_{el}&=0 \tag{10c}\\ 0<\rho_{\min}\le\rho_e&\le1. \tag{10d} \end{aligned} \]

\[ K=\sum_{\mathrm{assemble}}k_e(\rho_e) \tag{11} \]

\[ C=d^\top Kd=\sum_e\rho_e^p d_e^\top K_0d_e \tag{12} \]

\[ D_\rho C=-d^\top K'd \tag{13} \]

\[ E(\rho_e)=\rho_e^pE_0 \tag{14} \]

\[ \frac{\partial k_e}{\partial\rho_e}=p\rho_e^{p-1}k_0 \tag{15} \]

\[ E(\rho_e)=\frac{\rho_e}{1+q(1-\rho_e)}E_0 \tag{16} \]

\[ \frac{\partial k_e}{\partial\rho_e} = \frac{1+q}{[1+q(1-\rho_e)]^2}k_0. \tag{17} \]

Level-set TO va topologik sezgirlik

\[ \psi(x,y)=\mathbf1_D(x,y) \cos\left(\frac{n_x\pi x}{l_x}\right) \cos\left(\frac{n_y\pi y}{l_y}\right) \tag{18} \]

\[ T(p)= \lim_{\epsilon\to0^+} \frac{\phi(\Omega_\epsilon)-\phi(\Omega)} {\pi\epsilon^2} \tag{19} \]

\[ \Omega_\tau= \{p_\tau\in\mathbb R^2\mid p_\tau=p+\tau n,\ p\in\Omega_\epsilon\} \tag{20} \]

\[ T(p)= \frac{4}{1+\nu}\sigma:\epsilon - \frac{1-3\nu}{1-\nu^2} tr(\sigma)tr(\epsilon). \tag{21} \]

Modified HJE matnida \(g=-sign(\psi)T\) ko‘rsatilgan, biroq shu gapda ijobiy \(w\) og‘irlik koeffitsiyenti tilga olinadi. Ko‘rsatilgan formulada \(w\) yo‘q; manbadagi bu nomuvofiqlik tuzatilmagan.

ESO, BESO va PareTO

\[ \Omega_\tau=\{p\mid D\phi(p)>\tau\} \tag{22} \]

\[ \Omega_{\tau^-}^{keep} =\{p\in\Omega\mid\alpha(p)>\tau^-\} \tag{23} \]

\[ \Omega_{\tau^+}^{add} =\{p\in\Omega\mid\alpha(p)>\tau^+\} \tag{24} \]

\[ \begin{aligned} \min_{\Omega(x)\subseteq D}\;&\{V,\phi(d)\}\\ V(x)&\le V^*,\\ R_{el}(d)&=0. \end{aligned} \tag{25} \]

\[ \Delta V(\Omega_1,\Omega_2) = V(\Omega_1\setminus\Omega_2)+ V(\Omega_2\setminus\Omega_1) \le\delta. \tag{26} \]

Ishlab chiqarish filtrlari

\[ \chi_e= \begin{cases} 1,&e\in\Omega_{design}\\ 0,&\text{aks holda} \end{cases} \tag{27} \]

\[ \tilde\rho_e= \frac{\sum_iH_{ei}\chi_i\rho_i}{\sum_iH_{ei}\chi_i} \tag{28} \]

\[ H_{ei}= \begin{cases} r_{\min}-dist(e,i),&dist(e,i)\le r_{\min}\\ 0,&\text{aks holda} \end{cases} \tag{29} \]

\[ \frac{\partial\phi}{\partial\rho_i} = \sum_e \frac{\partial\phi}{\partial\tilde\rho_e} \frac{\partial\tilde\rho_e}{\partial\rho_i} \tag{30} \]

\[ \frac{\partial\tilde\rho_e}{\partial\rho_i} = \frac{H_{ei}\chi_i}{\sum_jH_{ej}\chi_j} \tag{31} \]

\[ \widehat{\left(\frac{\partial\phi}{\partial\rho_i}\right)} = \frac{1}{\max(\rho_i,\epsilon)} \sum_eH_{ei}\rho_i \frac{\partial\phi}{\partial\tilde\rho_e}, \quad\epsilon=10^{-3} \tag{32} \]

\[ \hat\rho_e= \frac{\tanh(\beta\eta)+\tanh[\beta(\rho_e-\eta)]} {\tanh(\beta\eta)+\tanh[\beta(1-\eta)]} \tag{33} \]

\[ \frac{\partial\hat\rho_e}{\partial\rho_e} = \frac{\beta[1-\tanh^2(\beta(\rho_e-\eta))]} {\tanh(\beta\eta)+\tanh[\beta(1-\eta)]} \tag{34} \]

\[ \frac{\partial\phi}{\partial\rho_e} = \frac{\partial\phi}{\partial\hat\rho_e} \frac{\partial\hat\rho_e}{\partial\rho_e} \tag{35} \]

\[ \frac{\partial\phi}{\partial\rho_i} = \sum_e \frac{\partial\phi}{\partial\hat\rho_e} \frac{\partial\hat\rho_e}{\partial\tilde\rho_e} \frac{\partial\tilde\rho_e}{\partial\rho_i}. \tag{36} \]

Retain cheklovi

\[ \rho_e^{filtered}=1 \quad\text{agar}\quad\chi_e^{ret}=1 \tag{37} \]

\[ \left.\frac{\partial\phi}{\partial\rho_e}\right|_{\chi_e^{ret}=1} = \min_i\left(\frac{\partial\phi}{\partial\rho_i}\right) \tag{38} \]

\[ \left.\frac{\partial\phi}{\partial\phi_e}\right|_{\chi_e^{ret}=1}=0 \tag{39} \]

\[ \left.\frac{\partial\phi}{\partial\rho_e}\right|_{\chi_e^{ret}=1} = \max_i\left(\frac{\partial\phi}{\partial\rho_i}\right). \tag{40} \]

Ko‘p yuk, o‘z og‘irligi va issiqlik

\[ C_{avg} = \frac1{N_L}\sum_{\ell=1}^{N_L}F^{(\ell)T}d^{(\ell)} \tag{41} \]

\[ \begin{aligned} \min_\rho\;&\phi(d;\rho) \tag{42a}\\ \sum_e\rho_ev_e-V^*&\le0 \tag{42b}\\ K_{el}(\rho)d-f_{el}-b(\rho)&=0 \tag{42c}\\ 0<\rho_{\min}\le\rho_e&\le1. \tag{42d} \end{aligned} \]

\[ \rho_e^{eff}=\rho_e\rho_e^{mat} \tag{43} \]

\[ b=\sum_e\rho_e^{eff}a_ev_e \tag{44} \]

\[ b'=\rho_e^{mat}a_ev_e. \tag{45} \]

\[ \begin{aligned} \nabla\cdot(\kappa\nabla T)+f&=0 \tag{46a}\\ T-T_0&=0 \tag{46b}\\ (\kappa\nabla T)\cdot n-q_n(x)&=0. \tag{46c} \end{aligned} \]

\[ \begin{aligned} \min_\rho\;&T^\top K_{th}(\rho)T \tag{47a}\\ \sum_e\rho_ev_e-V^*&\le0 \tag{47b}\\ K_{th}(\rho)T-f_{th}&=0 \tag{47c}\\ 0<\rho_{\min}\le\rho_e&\le1. \tag{47d} \end{aligned} \]

Stress va kengaytmalar

\[ \tilde\sigma_{vm,e}=\rho_e^{q_{vm}}\sigma_{vm,e} \tag{48} \]

\[ \sigma_{PN} = \left(\sum_e\tilde\sigma_{vm,e}^{p_{vm}}\right)^{1/p_{vm}} \tag{49} \]

\[ \begin{aligned} \min_\rho\;&\sigma_{PN}(d;\rho) \tag{50a}\\ \sum_e\rho_ev_e-V^*&\le0 \tag{50b}\\ R_{el}(d;\rho)&=0 \tag{50c}\\ 0<\rho_{\min}\le\rho_e&\le1. \tag{50d} \end{aligned} \]

\[ \frac{d\sigma_{PN}}{d\rho_e} = \frac{\partial\sigma_{PN}}{\partial\hat\rho_e} \frac{\partial\hat\rho_e}{\partial\bar\rho_e} \frac{\partial\bar\rho_e}{\partial\rho_e} \tag{51} \]

\[ \frac{d\sigma_{PN}}{d\rho} = \frac{\partial\sigma_{PN}}{\partial\rho} + \lambda^\top\frac{\partial F}{\partial\rho} \tag{52} \]

\[ K(\rho)^\top\lambda = \frac{\partial\sigma_{PN}}{\partial d}. \tag{53} \]

Stress misolida 6.000 element, SIMP p=3, E=100 GPa, ν=0.3 va \(p_{vm}=6\); maksimal von Mises 60,6 MPa dan 23,0 MPa ga kamayadi. Lokal hajm beam matnida p=6, kodda localPNorm=16. Gripper: 80.000 aktiv element, lokal ulush 0,65, radius 10, matnda p=6.

Suyuqlik kengaytmasi steady laminar incompressible Navier–Stokes va Brinkman penalty ishlatadi. Pipe bend/double pipe: 2.000 element, SIMP p=3, MMA, \(V_f=0.3\), \(U_{in}=1\) m/s, \(\rho=1\) kg/m³, \(\mu=1\) kg/(m·s). Wind tunnel: 40.000 element, Re=10, aktiv markaz [1.35,0.5], w=1,5 m, h=0,5 m; dissipatsiya, drag va lift maqsadlari.

O‘zbekiston nuqtai nazaridan

Ish O‘zbekiston ma’lumotlariga asoslanmagan va bu yerda sanoat miqyosida tasdiqlanmagan. Biroq ochiq MATLAB kodi, FEA, umumiy 2D geometriya, ishlab chiqarish cheklovlari va issiqlik/suyuqlik kengaytmalari O‘zbekistondagi mexanika, mashinasozlik va oliy muhandislik ta’limiga moslashtiriladigan platforma bo‘lishi mumkin. Mahalliy material parametrlari, ishlab chiqarish toleranslari, hisoblash infratuzilmasi va eksperimental validatsiya alohida tekshirilishi lozim.

Tadqiqot Usuli va Natijalari

Usul / misol	C (N·m)	δmax (m)	σvm,max (MPa)
LSSO Cantilever past	7.85	9.37e-05	10.5
LSSO Cantilever o‘rta	12.5	9.99e-05	4.93
LSSO L-bracket yuqori	25.8	2.76e-04	37.8
LSSO L-bracket o‘rta	25.7	2.89e-04	54.3
LSSO MBB	22.7	2.48e-04	6.18
SIMP-OC Cantilever past	6.98	7.46e-05	6.94
SIMP-OC Cantilever o‘rta	10.6	8.50e-05	8.83
SIMP-OC L-bracket yuqori	22.9	2.46e-04	21.0
SIMP-OC L-bracket o‘rta	21.7	2.42e-04	23.0
SIMP-OC MBB	15.0	1.64e-04	6.98
Standard HJE Cantilever past	6.38	6.85e-05	2.34
Standard HJE Cantilever o‘rta	9.68	7.84e-05	2.83
Standard HJE L-bracket yuqori	21.2	2.27e-04	25.2
Standard HJE L-bracket o‘rta	19.7	2.21e-04	36.1
Standard HJE MBB	14.0	1.53e-04	3.98
Modified HJE Cantilever past	6.31	6.70e-05	2.59
Modified HJE Cantilever o‘rta	9.65	7.76e-05	3.06
Modified HJE L-bracket yuqori	20.1	2.20e-04	9.49
Modified HJE L-bracket o‘rta	19.3	2.19e-04	29.5
Modified HJE MBB	13.7	1.50e-04	4.74

Verianla Live: ESO, BESO va PareTO

Manba Table 6 qiymatlari.

Misol / usul	C (N·m)	δmax (m)	σvm,max (MPa)	Source
Cantilever past — ESO	6.48	6.89e-05	4.52	Table 6
Cantilever past — BESO	6.43	6.86e-05	5.79	Table 6
Cantilever past — PareTO	6.32	6.74e-05	3.24	Table 6
Cantilever o‘rta — ESO	9.76	7.83e-05	6.69	Table 6
Cantilever o‘rta — BESO	9.76	7.83e-05	6.69	Table 6
Cantilever o‘rta — PareTO	9.59	7.69e-05	3.28	Table 6
L-bracket yuqori — ESO	20.4	2.23e-04	7.99	Table 6
L-bracket yuqori — BESO	20.4	2.23e-04	7.99	Table 6
L-bracket yuqori — PareTO	20.3	2.21e-04	8.01	Table 6
L-bracket o‘rta — ESO	21.2	3.20e-04	490	Table 6
L-bracket o‘rta — BESO	21.2	3.20e-04	490	Table 6
L-bracket o‘rta — PareTO	19.2	2.16e-04	19.6	Table 6
MBB — ESO	14.2	1.55e-04	19.4	Table 6
MBB — BESO	14.2	1.55e-04	19.4	Table 6
MBB — PareTO	13.4	1.46e-04	5.39	Table 6

3D bosilgan gripper compliance: LSSO \(1.04\times10^{-5}\), standard HJE \(8.49\times10^{-6}\), modified HJE \(8.33\times10^{-6}\), SIMP \(8.66\times10^{-6}\), PareTO \(8.29\times10^{-6}\) N·m. Bu umumiy metod reytingi emas.

Manba va Usul Bo‘yicha Izoh

Asl nom: STORX: An Open-Source Object-Oriented Framework for Shape and Topology Optimization in MATLAB.

Mualliflar: Amir M. Mirzendehdel; Krishnan Suresh.

Manba: arXiv:2606.17291v2, cs.CE, 25-iyul 2026.

DOI: 10.48550/arXiv.2606.17291.

Litsenziya: CC BY 4.0.

Hakamlilik: Ushbu ish hakam ko‘rigidan o‘tmagan preprint; natijalarni shu nashr bosqichi hisobga olinib baholash lozim.

Kod: https://github.com/DEL-KU/storx.

Manfaatlar to‘qnashuvi: Mualliflar to‘qnashuv yo‘qligini bildiradi.

Generativ AI: Qo‘lyozmaning ayrim qismlarini ravshan va o‘qilishi oson qilish uchun ishlatilgan; texnik mazmun va natijalar mualliflar tomonidan yaratilgan va tekshirilgan.

Cheklovlar: Asosiy framework 2Dga yo‘naltirilgan; benchmark natijalari metodlarning universal ustunligini isbotlamaydi; OOP abstraksiyasi maxsus kompakt kodlarga qaraganda runtime overhead keltiradi.

Manba ichidagi nomuvofiqlik: Lokal hajm p=6 / localPNorm=16, modified-HJE w omili va wind-tunnel aktiv maydon ifodasi muammolari o‘zgartirilmasdan saqlandi.

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Ulashish:

STORX: MATLAB’da shakl va topologiya optimallashtirishini birlashtiruvchi ochiq manbali obyektga yo‘naltirilgan platforma