Octree-based adaptive mesh refinement and the shifted boundary method for efficient fluid dynamics simulations

Cheng-Hau Yang; Guglielmo Scovazzi; Adarsh Krishnamurthy; Baskar Ganapathysubramanian

doi:10.3934/acse.2025012

Article Contents

June 2025, Volume 4, 57-84. Doi: 10.3934/acse.2025012

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Octree-based adaptive mesh refinement and the shifted boundary method for efficient fluid dynamics simulations

1.
Department of Mechanical Engineering, Iowa State University, Ames, IA 50011, USA
2.
Department of Civil and Environmental Engineering, Duke University, Durham, NC 27708, USA

^*Corresponding author: Baskar Ganapathysubramanian
^*Corresponding author: Baskar Ganapathysubramanian

Received: December 2024

Revised: March 2025

Published online: April 15, 2025

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Abstract

This paper presents an adaptive mesh refinement (AMR) framework integrated with the shifted boundary method (SBM) for incompressible flow and coupled thermal-flow simulations. Our framework leverages octree-based AMR, enabling hierarchical and dynamic mesh refinement driven by vorticity magnitude. This strategy enables capturing complex vorticity structures and steep thermal gradients while significantly reducing computational costs compared to traditional uniform refinement approaches, particularly for flows around complex geometries. The octree-based architecture ensures efficient data management, including robust intergrid transfer and load balancing, which is critical for scalability in distributed-memory environments. Dynamic mesh adaptivity is demonstrated for complex geometries where achieving ideal refinement is often non-trivial due to the irregular boundaries. SBM enhances this adaptability by accurately enforcing boundary conditions on intricate and non-conformal geometries without requiring boundary-fitted meshes. Together, these methods address longstanding challenges in computational fluid dynamics, providing a resource-efficient yet accurate approach for capturing critical flow and thermal features. The utility of the framework is demonstrated through numerical experiments, showcasing its ability to adapt dynamically to evolving flow and thermal patterns in diverse and challenging geometries.

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Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.

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Figure 1. The surrogate domain, its boundary, and the distance vector $ \boldsymbol{d} $

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Figure 2. Flow past a two-dimensional circular cylinder at $ Re = 100 $: mesh with AMR vorticity contours are shown in the top pane, while mesh refinement levels are shown in the bottom pane. The figure highlights the refined mesh regions around the cylinder due to AMR, which optimally reduces mesh elements while retaining flow features

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Figure 3. Flow past a two-dimensional circular cylinder at $ Re = 100 $: mesh with wake refinement (refinement level 12, mesh size $ 30 \times 2^{-12} $)

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Figure 4. Flow past a two-dimensional circular cylinder at $ Re = 100 $: history of the number of mesh nodes for AMR versus a regular grid with uniform mesh refinement in the wake region (indicated by the white box in Figure 2b). The significant reduction in node count achieved by AMR emphasizes computational efficiency

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Figure 5. Lid-driven cavity flow with complex internal geometries: velocity profiles. The first row of pictures displays an overlay of the vorticity with the AMR mesh, while the second row displays velocity profiles around a cat-shaped obstacle, comparing results between the SBM and a BFM. These simulations, conducted at Reynolds numbers of $Re = 50 $, $Re = 500 $, and $Re = 1000 $, illustrate how both the obstacle shape and Reynolds number affect the flow dynamics

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Figure 6. Lid-driven cavity flow with complex internal geometries: Plot of the y-direction velocity ($ v $) along $ y = 0.5 $, with $ x $ ranging from 0.06 to 0.2 inside the flow chamber. Comparison of AMR-SBM and SBM at different uniform refinement levels. AMR-SBM refers to simulations using dynamic AMR with SBM, while SBM represents simulations with uniform meshes. AMR-SBM outperforms most SBM simulations with uniform meshes, except for the case with a uniform mesh refinement level of $ 9 $

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Figure 7. Mixed convection inside a lid-driven cavity: Velocity magnitude distribution (first row of plots) and temperature distribution (second row of plots), for varying Richardson numbers ($ Ri $). These plots demonstrates how changes in $ Ri $ affect the temperature field, with the AMR approach capturing all essential temperature gradients and flow patterns

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Figure 8. Mixed convection inside a lid-driven cavity: temperature distribution for varying Richardson numbers ($ Ri $). The figure demonstrates how changes in $ Ri $ affect the temperature field, with the AMR approach capturing essential temperature gradients

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Figure 9. Mixed convection inside a lid-driven cavity: Temperature distribution for different Richardson numbers ($ Ri $). This figure compares results obtained using the SBM with AMR and uniform meshes. The term AMR-SBM refers to simulations performed with dynamic AMR and SBM, whereas SBM denotes simulations using SBM with uniform octree meshes

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Figure 10. Natural convection past a star-shaped domain: vorticity contours over time. The vorticity-based AMR allows for accurate tracking of vorticity at different non-dimensional times

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Figure 11. Natural convection past a star-shaped domain: temperature contours at various non-dimensional times. The AMR captures well the evolution of high thermal gradients over time

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Figure 12. Visualization of the initial mesh refinement setup for the flow past a sphere, highlighting the distribution before adaptive refinement is introduced

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Figure 13. Flow past a sphere at $ Re = 300 $. (a) and (b) illustrate octree element volumes and the Q-criterion respectively, while (c) compares the centerline average streamwise velocity profile with benchmark data [84]

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Table 1. Flow past a two-dimensional circular cylinder at $ Re = 100 $: comparison of drag coefficient ($ C_d $) and Strouhal number ($ St $). The results of SBM with AMR are consistent with the literature, confirming the accuracy of the proposed approach for different mesh refinement levels

Study	$ C_d $	$ St $
Liu et al. [52]	1.350	0.1650
Lai et al. [49]	1.447	0.1650
Uhlmann [85]	1.453	0.1690
Yang et al. [98]	1.393	0.1650
Kamensky et al. [40]	1.386	0.1700
Current (finest element size = $ 30/2^{12} $)	1.401	0.1709
Current (finest element size = $ 30/2^{13} $)	1.404	0.1707
Current (finest element size = $ 30/2^{14} $)	1.405	0.1707

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Table 2. Flow past a two-dimensional circular cylinder at $ Re = 100 $: Comparison of the drag coefficient ($ C_d $) and Strouhal number ($ St $). The term AMR-SBM refers to simulations conducted with dynamic AMR and the SBM, while SBM denotes simulations using SBM with local mesh refinement. Both AMR-SBM and SBM simulations are performed with a base refinement level of 5 and a boundary layer refinement level of 12 near the circular disk. For SBM (without AMR), we locally refine the region up to $ level_{wake} $ just upstream of the circular disk and throughout the wake region, as shown by the white box in Figure 2b. Below, we present SBM simulations with $ level_{wake} $ set to 10, 11, and 12, and compare the results with AMR-SBM. We treat the SBM simulation with the highest refinement level in the upstream and wake region as the ground truth and use it to calculate $ Error_{C_d} = \frac{|C_d - C_d^{\text{ground truth}|}}{|C_d^{\text{ground truth}}|} $ and $ Error_{St} = \frac{|St - St^{\text{ground truth}}|}{|St^{\text{ground truth}}|} $

Study	Highest Mesh Nodes	$ C_d $	$ St $	$ Error_{C_d} $	$ Error_{St} $
AMR-SBM	75464	1.401	0.1709	0.214%	0.0%
SBM ($ level_{wake} = 9 $)	29053	1.507	0.1685	7.336%	1.404%
SBM ($ level_{wake} = 10 $)	101694	1.411	0.1717	0.499%	0.468%
SBM ($ level_{wake} = 11 $)	389987	1.410	0.1717	0.427%	0.468%
SBM ($ level_{wake} = 12 $)	1544191	1.404	0.1709	0.0%	0.0%

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Table 3. Lid-driven cavity flow with complex internal geometries: grid size comparison (in terms of mesh nodes) for uniform mesh versus AMR

Reynolds Number	Uniform Mesh	AMR Mesh	Compression Ratio
50	238100	34019	6.99
500		34487	6.90
1000		34676	6.87

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Table 4. Mixed convection inside a lid-driven cavity: grid size comparison (in terms of mesh nodes) for uniform mesh versus AMR

Richardson number	Uniform Mesh	AMR Mesh	Compression Ratio
0.01	14780	4252	3.48
1		5237	2.82
5		9430	1.57

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Table 5. Flow past a sphere at $ Re = 300 $: comparison of the drag coefficient ($ C_d $) and Strouhal number ($ St $) against various references. The proposed combined SBM-AMR approach compares well against previous experimental and computational studies

Study	Cd	St
Roos and Willmarth [69] (interpolated experiment value)	0.629	-
Le Clair et al. [51]	0.632	-
Johnson and Patel [38]	0.656	0.137
Marella et al. [57]	0.621	0.133
Vanella et al. [86]	0.634	0.132
Wang and Zhang [89]	0.680	0.135
Angelidis et al. [2]	0.665	0.132
Kang et al. [42]	0.663	0.134
Current	0.622	0.134

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References

[1]	A. S. Almgren, J. B. Bell, P. Colella, L. H. Howell and M. L. Welcome, A conservative adaptive projection method for the variable density incompressible navier–stokes equations, Journal of computational Physics, 142 (1998), 1-46. doi: 10.1006/jcph.1998.5890.
[2]	D. Angelidis, S. Chawdhary and F. Sotiropoulos, Unstructured cartesian refinement with sharp interface immersed boundary method for 3D unsteady incompressible flows, Journal of Computational Physics, 325 (2016), 272-300. doi: 10.1016/j.jcp.2016.08.028.
[3]	A. Ankit, J. Zhang, S. Eisenträger and C. Song, An octree pattern-based massively parallel pcg solver for elasto-static and dynamic problems, Computer Methods in Applied Mechanics and Engineering, 404 (2023), 115779. doi: 10.1016/j.cma.2022.115779.
[4]	N. Antonelli, R. Aristio Andrea Gorgi, R. Zorrilla, R. Rossi, G. Scovazzi and R. Wüchner, The shifted boundary method in isogeometric analysis, Computer Methods in Applied Mechanics and Engineering, 430 (2024), 117228. doi: 10.1016/j.cma.2024.117228.
[5]	N. Atallah, C. Canuto and G. Scovazzi, Analysis of the shifted boundary method for the poisson problem in domains with corners, Mathematics of Computation, 90 (2021), 2041-2069. doi: 10.1090/mcom/3641.
[6]	N. Atallah, C. Canuto and G. Scovazzi, The shifted boundary method for solid mechanics, International Journal for Numerical Methods in Engineering, 122 (2021), 5935-5970. doi: 10.1002/nme.6779.
[7]	N. M. Atallah, C. Canuto and G. Scovazzi, The second-generation shifted boundary method and its numerical analysis, Computer Methods in Applied Mechanics and Engineering, 372 (2020), 113341. doi: 10.1016/j.cma.2020.113341.
[8]	N. M. Atallah, C. Canuto and G. Scovazzi, The high-order shifted boundary method and its analysis, Computer Methods in Applied Mechanics and Engineering, 394 (2022), 114885. doi: 10.1016/j.cma.2022.114885.
[9]	J. Baiges, R. Codina, A. Pont and E. Castillo, An adaptive fixed-mesh ale method for free surface flows, Computer Methods in Applied Mechanics and Engineering, 313 (2017), 159-188. doi: 10.1016/j.cma.2016.09.041.
[10]	A. Balu, M. R. Rajanna, J. Khristy, F. Xu, A. Krishnamurthy and M.-C. Hsu, Direct immersogeometric fluid flow and heat transfer analysis of objects represented by point clouds, Computer Methods in Applied Mechanics and Engineering, 404 (2023), 115742. doi: 10.1016/j.cma.2022.115742.
[11]	M. J. Berger and P. Colella, Local adaptive mesh refinement for shock hydrodynamics, Journal of computational Physics, 82 (1989), 64-84. doi: 10.1016/0021-9991(89)90035-1.
[12]	M. J. Berger and J. Oliger, Adaptive mesh refinement for hyperbolic partial differential equations, Journal of Computational Physics, 53 (1984), 484-512. doi: 10.1016/0021-9991(84)90073-1.
[13]	C. Bertoglio and A. Caiazzo, A tangential regularization method for backflow stabilization in hemodynamics, Journal of Computational Physics, 261 (2014), 162-171. doi: 10.1016/j.jcp.2013.12.057.
[14]	J. Bielak, O. Ghattas and E. J. Kim, Parallel octree-based finite element method for large-scale earthquake ground motion simulation, Computer Modeling in Engineering and Sciences, 10 (2005), 99-112. doi: 10.3970/cmes.2005.010.099.
[15]	I. Borazjani, L. Ge and F. Sotiropoulos, Curvilinear immersed boundary method for simulating fluid structure interaction with complex 3D rigid bodies, Journal of Computational Physics, 227 (2008), 7587-7620. doi: 10.1016/j.jcp.2008.04.028.
[16]	M. Braack and P. Boguslaw Mucha, Directional do-nothing condition for the Navier-Stokes equations, Journal of Computational Mathematics, 32 (2014), 507-521. doi: 10.4208/jcm.1405-m4347.
[17]	G. L. Bryan, M. L. Norman, B. W. O'Shea, T. Abel, J. H. Wise, M. J. Turk, D. R. Reynolds, D. C. Collins, P. Wang, S. W. Skillman and et al, Enzo: An adaptive mesh refinement code for astrophysics, The Astrophysical Journal Supplement Series, 211 (2014), 19. doi: 10.1088/0067-0049/211/2/19.
[18]	Z. Chen, C. Shu, L. Yang, X. Zhao and N. Liu, Immersed boundary–simplified thermal lattice Boltzmann method for incompressible thermal flows, Physics of Fluids, 32 (2020), 01. doi: 10.1063/1.5138711.
[19]	J. Cheng, F. Yu and L. T. Zhang, OpenIFEM: A high performance modular open-source software of the immersed finite element method for fluid-structure interactions, Computer Modeling in Engineering and Sciences: CMES, 119 (2019), 91. doi: 10.32604/cmes.2019.04318.
[20]	W. J. Coirier, An adaptively-refined, Cartesian, Cell-based Scheme for the Euler and Navier-Stokes Equations, University of Michigan, 1994.
[21]	P. Colella, D. T. Graves, T. J. Ligocki, D. F. Martin, D. Modiano, D. B. Serafini and B. Van Straalen, Chombo software package for amr applications design document, Available at the Chombo Website: http://seesar.lbl.gov/ANAG/chombo/(September 2008), 2 (2009). doi: 10.6084/m9.figshare.28599755.v1.
[22]	J. H. Collins, A. Lozinski and G. Scovazzi, A penalty-free shifted boundary method of arbitrary order, Computer Methods in Applied Mechanics and Engineering, 417 (2023), 116301. doi: 10.1016/j.cma.2023.116301.
[23]	O. Colomés, A. Main, L. Nouveau and G. Scovazzi, A weighted shifted boundary method for free surface flow problems, Journal of Computational Physics, 424 (2021), 109837. doi: 10.1016/j.jcp.2020.109837.
[24]	T. Colonius and K. Taira, A fast immersed boundary method using a nullspace approach and multi-domain far-field boundary conditions, Computer Methods in Applied Mechanics and Engineering, 197 (2008), 2131-2146. doi: 10.1016/j.cma.2007.08.014.
[25]	M. S. Day and J. B. Bell, Numerical simulation of laminar reacting flows with complexchemistry, Combustion Theory and Modelling, 4 (2000), 535. doi: 10.1088/1364-7830/4/4/309.
[26]	F. de Prenter, C. V. Verhoosel and E. H. van Brummelen, Preconditioning immersed isogeometric finite element methods with application to flow problems, Computer Methods in Applied Mechanics and Engineering, 348 (2019), 604-631. doi: 10.1016/j.cma.2019.01.030.
[27]	F. de Prenter, C. V. Verhoosel, G. J. van Zwieten and E. H. van Brummelen, Condition number analysis and preconditioning of the finite cell method, Computer Methods in Applied Mechanics and Engineering, 316 (2017), 297-327. doi: 10.1016/j.cma.2016.07.006.
[28]	S. Dong, A convective-like energy-stable open boundary condition for simulations of incompressible flows, Journal of Computational Physics, 302 (2015), 300-328. doi: 10.1016/j.jcp.2015.09.017.
[29]	A. Düster, J. Parvizian, Z. Yang and E. Rank, The finite cell method for three-dimensional problems of solid mechanics, Computer Methods in Applied Mechanics and Engineering, 197 (2008), 3768-3782. doi: 10.1016/j.cma.2008.02.036.
[30]	M. Esmily Moghadam, Y. Bazilevs, T.-Y. Hsia, I. Vignon-Clementel, A. Marsden and M. (MOCHA, A comparison of outlet boundary treatments for prevention of backflow divergence with relevance to blood flow simulations, Computational Mechanics, 48 (2011), 277-291. doi: 10.1007/s00466-011-0599-0.
[31]	M. Feistauer and T. Neustupa, On the existence of a weak solution of viscous incompressible flow past a cascade of profiles with an arbitrarily large inflow, Journal of Mathematical Fluid Mechanics, 15 (2013), 701-715. doi: 10.1007/s00021-013-0135-4.
[32]	D. Gamdha, K. Saurabh, B. Ganapathysubramanian and A. Krishnamurthy, Geometric modeling and physics simulation framework for building a digital twin of extrusion-based additive manufacturing, preprint, arXiv: 2305.07120, 2023.
[33]	L. Ge. and F. Sotiropoulos, A numerical method for solving the 3D unsteady incompressible Navier-Stokes equations in curvilinear domains with complex immersed boundaries, Journal of Computational Physics, 225 (2007), 1782-1809. doi: 10.1016/j.jcp.2007.02.017.
[34]	E. Heisler, C. Yang, A. Deshmukh, B. Ganapathysubramanian and H. Sundar, Generating finite element codes combining adaptive octrees with complex geometries, preprint, arXiv: 2305.19398, 2023.
[35]	T. Hoang, C. V. Verhoosel, C. Qin, F. Auricchio, A. Reali and E. H. van Brummelen, Skeleton-stabilized immersogeometric analysis for incompressible viscous flow problems, Computer Methods in Applied Mechanics and Engineering, 344 (2019), 421-450. doi: 10.1016/j.cma.2018.10.015.
[36]	M. Ishii, M. Fernando, K. Saurabh, B. Khara, B. Ganapathysubramanian and H. Sundar, Solving pdes in space-time: 4d tree-based adaptivity, mesh-free and matrix-free approaches, In Proceedings of the International Conference for High Performance Computing, Networking, Storage and Analysis, (2019), 1-61. doi: 10.1145/3295500.3356198.
[37]	M. Ismail, V. Gravemeier, A. Comerford and W. Wall, A stable approach for coupling multidimensional cardiovascular and pulmonary networks based on a novel pressure-flow rate or pressure-only neumann boundary condition formulation, International Journal for Numerical Methods in Biomedical Engineering, 30 (2014), 447-469. doi: 10.1002/cnm.2611.
[38]	T. A. Johnson and V. C. Patel, Flow past a sphere up to a Reynolds number of 300, Journal of Fluid Mechanics, 378 (1999), 19-70. doi: 10.1017/S0022112098003206.
[39]	J. Jomo, O. Oztoprak, F. de Prenter, N. Zander, S. Kollmannsberger and E. Rank, Hierarchical multigrid approaches for the finite cell method on uniform and multi-level hp-refined grids, Computer Methods in Applied Mechanics and Engineering, 386 (2021), 114075. doi: 10.1016/j.cma.2021.114075.
[40]	D. Kamensky, M.-C. Hsu, D. Schillinger, J. A. Evans, A. Aggarwal, Y. Bazilevs, M. S. Sacks and T. J. R. Hughes, An immersogeometric variational framework for fluid-structure interaction: Application to bioprosthetic heart valves, Computer Methods in Applied Mechanics and Engineering, 284 (2015), 1005-1053. doi: 10.1016/j.cma.2014.10.040.
[41]	D. Kamensky, Open-source immersogeometric analysis of fluid-structure interaction using fenics and tigar, Computers and Mathematics with Applications, 81 (2021), 634-648. doi: 10.1016/j.camwa.2020.01.023.
[42]	S. Kang and A. Masud, A variational multiscale method with immersed boundary conditions for incompressible flows, Meccanica, 56 (2021), 1397-1422. doi: 10.1007/s11012-020-01227-w.
[43]	E. N. Karatzas, G. Stabile, L. Nouveau, G. Scovazzi and G. Rozza, A reduced-order shifted boundary method for parametrized incompressible Navier-Stokes equations, Computer Methods in Applied Mechanics and Engineering, 370 (2020), 113273. doi: 10.1016/j.cma.2020.113273.
[44]	K. Khanafer and S. M. Aithal, Laminar mixed convection flow and heat transfer characteristics in a lid driven cavity with a circular cylinder, International Journal of Heat and Mass Transfer, 66 (2013), 200-209. doi: 10.1016/j.ijheatmasstransfer.2013.07.023.
[45]	M. A. Khanwale, A. D. Lofquist, H. Sundar, J. A. Rossmanith and B. Ganapathysubramanian, Simulating two-phase flows with thermodynamically consistent energy stable cahn-hilliard Navier-Stokes equations on parallel adaptive octree based meshes, Journal of Computational Physics, 419 (2020), 109674. doi: 10.1016/j.jcp.2020.109674.
[46]	M. A. Khanwale, K. Saurabh, M. Fernando, V. M. Calo, H. Sundar, J. A. Rossmanith and B. Ganapathysubramanian, A fully-coupled framework for solving cahn-hilliard navier-stokes equations: Second-order, energy-stable numerical methods on adaptive octree based meshes, Computer Physics Communications, 280 (2022), 108501. doi: 10.1016/j.cpc.2022.108501.
[47]	A. M. Khokhlov, Fully threaded tree algorithms for adaptive refinement fluid dynamics simulations, Journal of Computational Physics, 143 (1998), 519-543. doi: 10.1006/jcph.1998.9998.
[48]	S. Kim, K. Saurabh, M. A. Khanwale, A. Mani, R. K. Anand and B. Ganapathysubramanian, Direct numerical simulation of electrokinetic transport phenomena in fluids: Variational multi-scale stabilization and octree-based mesh refinement, Journal of Computational Physics, 500 (2024), 112747, 25 pp. doi: 10.1016/j.jcp.2023.112747.
[49]	M. Lai and C. S. Peskin, An immersed boundary method with formal second-order accuracy and reduced numerical viscosity, Journal of computational Physics, 160 (2000), 705-719. doi: 10.1006/jcph.2000.6483.
[50]	M. Lanzendörfer and J. Stebel, On pressure boundary conditions for steady flows of incompressible fluids with pressure and shear rate dependent viscosities, Applications of Mathematics, 56 (2011), 265-285. doi: 10.1007/s10492-011-0016-1.
[51]	B. P. Le Clair, A. E. Hamielec and H. R. Pruppacher, A numerical study of the drag on a sphere at low and intermediate Reynolds numbers, Journal of Atmospheric Sciences, 27 (1970), 308-315. doi: 10.1175/1520-0469(1970)027<0308:ANSOTD>2.0.CO;2.
[52]	C. Liu, X. Zheng and C. H. Sung, Preconditioned multigrid methods for unsteady incompressible flows, Journal of Computational physics, 139 (1998), 35-57. doi: 10.1006/jcph.1997.5859.
[53]	W. K. Liu, Y. Liu, D. Farrell, L. Zhang, X. S. Wang, Y. Fukui, N. Patankar, Y. Zhang, C. Bajaj, J. Lee, et al., Immersed finite element method and its applications to biological systems, Computer methods in Applied Mechanics and Engineering, 195 (2006), 1722-1749. doi: 10.1016/j.cma.2005.05.049.
[54]	X. Liu, Z. Xie and S. Dong, On a simple and effective thermal open boundary condition for convective heat transfer problems, International Journal of Heat and Mass Transfer, 151 (2020), 119355. doi: 10.1016/j.ijheatmasstransfer.2020.119355.
[55]	A. Main and G. Scovazzi, The shifted boundary method for embedded domain computations. Part Ⅰ: Poisson and Stokes problems, Journal of Computational Physics, 372 (2018), 972-995. doi: 10.1016/j.jcp.2017.10.026.
[56]	A. Main and G. Scovazzi, The shifted boundary method for embedded domain computations. Part Ⅱ: Linear advection-diffusion and incompressible Navier-Stokes equations, J. Comput. Phys., 372 (2018), 996-1026. doi: 10.1016/j.jcp.2018.01.023.
[57]	S. Marella, S. Krishnan, H. Liu and H. Udaykumar, Sharp interface cartesian grid method i: An easily implemented technique for 3D moving boundary computations, Journal of Computational Physics, 210 (2005), 1-31. doi: 10.1016/j.jcp.2005.03.031.
[58]	D. J. R. Meagher, Octree Encoding: A New Technique for the Representation, Manipulation and Display of Arbitrary 3-d Objects by Computer, Electrical and Systems Engineering Department Rensseiaer Polytechnic …, 1980.
[59]	R. Mittal, H. Dong, M. Bozkurttas, F. M. Najjar, A. Vargas and A. Von Loebbecke, A versatile sharp interface immersed boundary method for incompressible flows with complex boundaries, Journal of Computational Physics, 227 (2008), 4825-4852. doi: 10.1016/j.jcp.2008.01.028.
[60]	R. Mittal and G. Iaccarino, Immersed boundary methods, Annual Review of Fluid Mechanics, 37 (2005), 239-261. doi: 10.1146/annurev.fluid.37.061903.175743.
[61]	S. Mohammadian, A. S. Kumar and C. Song, A multi-gpu based high-performance computing framework in elastodynamics simulation using octree meshes, Computer Methods in Applied Mechanics and Engineering, 436 (2025), 117723. doi: 10.1016/j.cma.2024.117723.
[62]	M. Nemec, M. Aftosmis and M. Wintzer, Adjoint-based adaptive mesh refinement for complex geometries, In 46th AIAA Aerospace Sciences Meeting and Exhibit, (2008), 725. doi: 10.2514/6.2008-725.
[63]	K. C. Ong and M.-C. Lai, An immersed boundary projection method for simulating the inextensible vesicle dynamics, Journal of Computational Physics, 408 (2020), 109277. doi: 10.1016/j.jcp.2020.109277.
[64]	J. Parvizian, A. Düster and E. Rank, Finite cell method: h- and p- extension for embedded domain methods in solid mechanics, Computational Mechanics, 41 (2007), 122-133. doi: 10.1007/s00466-007-0173-y.
[65]	C. S. Peskin, Flow patterns around heart valves: A numerical method, Journal of Computational Physics, 10 (1972), 252-271. doi: 10.1016/0021-9991(72)90065-4.
[66]	C. S. Peskin, The immersed boundary method, Acta Numerica, 11 (2002), 479-517. doi: 10.1017/S0962492902000077.
[67]	S. Popinet, Gerris: A tree-based adaptive solver for the incompressible euler equations in complex geometries, Journal of Computational Physics, 190 (2003), 572-600. doi: 10.1016/S0021-9991(03)00298-5.
[68]	S. Popinet, An accurate adaptive solver for surface-tension-driven interfacial flows, Journal of Computational Physics, 228 (2009), 5838-5866. doi: 10.1016/j.jcp.2009.04.042.
[69]	F. W. Roos and W. W. Willmarth, Some experimental results on sphere and disk drag, AIAA Journal, 9 (1971), 285-291. doi: 10.2514/3.6164.
[70]	R. S. Sampath and G. Biros, A parallel geometric multigrid method for finite elements on octree meshes, SIAM Journal on Scientific Computing, 32 (2010), 1361-1392. doi: 10.1137/090747774.
[71]	K. Saurabh, B. Gao, M. Fernando, S. Xu, M. A. Khanwale, B. Khara, M.-C. Hsu, A. Krishnamurthy, H. Sundar and B. Ganapathysubramanian, Industrial scale large eddy simulations with adaptive octree meshes using immersogeometric analysis, Computers and Mathematics with Applications, 97 (2021), 28-44. doi: 10.1016/j.camwa.2021.05.028.
[72]	K. Saurabh, M. Ishii, M. Fernando, B. Gao, K. Tan, M.-C. Hsu, A. Krishnamurthy, H. Sundar and B. Ganapathysubramanian, Scalable adaptive pde solvers in arbitrary domains, In Proceedings of the International Conference for High Performance Computing, Networking, Storage and Analysis, (2021), 1-15. doi: 10.1145/3458817.3476220.
[73]	K. Saurabh, M. Ishii, M. A. Khanwale, H. Sundar and B. Ganapathysubramanian, Scalable adaptive algorithms for next-generation multiphase flow simulations, In 2023 IEEE International Parallel and Distributed Processing Symposium (IPDPS), (2023), 590-601. doi: 10.1109/IPDPS54959.2023.00065.
[74]	D. Schillinger, Q. Cai, R.-P. Mundani and E. Rank, A review of the finite cell method for nonlinear structural analysis of complex CAD and image-based geometric models, In Advanced Computing, (2013), 1-23. doi: 10.1007/978-3-642-38762-3_1.
[75]	J. H. Seo and R. Mittal, A sharp-interface immersed boundary method with improved mass conservation and reduced spurious pressure oscillations, Journal of Computational Physics, 230 (2011), 7347-7363. doi: 10.1016/j.jcp.2011.06.003.
[76]	A. Stavrev, L. H. Nguyen, R. Shen, V. Varduhn, M. Behr, S. Elgeti and D. Schillinger, Geometrically accurate, efficient and flexible quadrature techniques for the tetrahedral finite cell method, Computer Methods in Applied Mechanics and Engineering, 310 (2016), 646-673. doi: 10.1016/j.cma.2016.07.041.
[77]	H. Sundar, G. Biros, C. Burstedde, J. Rudi, O. Ghattas and G. Stadler, Parallel geometric-algebraic multigrid on unstructured forests of octrees, In SC'12: Proceedings of the International Conference on High Performance Computing, Networking, Storage and Analysis, IEEE, (2012), 1-11. doi: 10.1109/SC.2012.91.
[78]	H. Sundar, R. S. Sampath and G. Biros, Bottom-up construction and 2: 1 balance refinement of linear octrees in parallel, SIAM Journal on Scientific Computing, 30 (2008), 2675-2708. doi: 10.1137/070681727.
[79]	M. Sussman and E. G. Puckett, A coupled level set and volume-of-fluid method for computing 3d and axisymmetric incompressible two-phase flows, Journal of Computational Physics, 162 (2000), 301-337. doi: 10.1006/jcph.2000.6537.
[80]	K. Taira, The Immersed Boundary Projection Method and Its Application to Simulation and Control of Flows Around Low-Aspect-Ratio Wings, California Institute of Technology, 2008. doi: 10.7907/VSDD-P465.
[81]	K. Taira and T. Colonius, The immersed boundary method: A projection approach, Journal of Computational Physics, 225 (2007), 2118-2137. doi: 10.1016/j.jcp.2007.03.005.
[82]	K. Tan, B. Gao, C.-H. Yang, E. L. Johnson, M.-C. Hsu, A. Passalacqua, A. Krishnamurthy and B. Ganapathysubramanian, A computational framework for transmission risk assessment of aerosolized particles in classrooms, Engineering with Computers, 40 (2024), 235-256. doi: 10.1007/s00366-022-01773-9.
[83]	T. E. Tezduyar, S. Sathe, R. Keedy and K. Stein, Space-time finite element techniques for computation of fluid-structure interactions, Computer Methods in Applied Mechanics and Engineering, 195 (2006), 2002-2027. doi: 10.1016/j.cma.2004.09.014.
[84]	A. G. Tomboulides and S. A. Orszag, Numerical investigation of transitional and weak turbulent flow past a sphere, Journal of Fluid Mechanics, 416 (2000), 45-73. doi: 10.1017/S0022112000008880.
[85]	M. Uhlmann, An immersed boundary method with direct forcing for the simulation of particulate flows, Journal of Computational Physics, 209 (2005), 448-476. doi: 10.1016/j.jcp.2005.03.017.
[86]	M. Vanella, P. Rabenold and E. Balaras, A direct-forcing embedded-boundary method with adaptive mesh refinement for fluid-structure interaction problems, Journal of Computational Physics, 229 (2010), 6427-6449. doi: 10.1016/j.jcp.2010.05.003.
[87]	R. Verzicco, Immersed boundary methods: Historical perspective and future outlook, Annual Review of Fluid Mechanics, 55 (2023), 129-155. doi: 10.1146/annurev-fluid-120720-022129.
[88]	C. Wang, F. Xu, M.-C. Hsu and A. Krishnamurthy, Rapid b-rep model preprocessing for immersogeometric analysis using analytic surfaces, Computer Aided Geometric Design, 52/53 (2017), 190-204. doi: 10.1016/j.cagd.2017.03.002.
[89]	S. Wang and X. Zhang, An immersed boundary method based on discrete stream function formulation for two-and three-dimensional incompressible flows, Journal of Computational Physics, 230 (2011), 3479-3499. doi: 10.1016/j.jcp.2011.01.045.
[90]	X. Wang and L. Zhang, Modified immersed finite element method for fully-coupled fluid-structure interactions, Computer Methods in Applied Mechanics and Engineering, 267 (2013), 150-169. doi: 10.1016/j.cma.2013.07.019.
[91]	D. Xu, O. Colomés, A. Main, K. Li, N. M. Atallah, N. Abboud and G. Scovazzi, A weighted shifted boundary method for immersed moving boundary simulations of stokes' flow, Journal of Computational Physics, 510 (2024), 113095. doi: 10.1016/j.jcp.2024.113095.
[92]	F. Xu, E. L. Johnson, C. Wang, A. Jafari, C. Yang, M. S. Sacks, A. Krishnamurthy and M. Hsu, Computational investigation of left ventricular hemodynamics following bioprosthetic aortic and mitral valve replacement, Mechanics Research Communications, 112 (2021), 103604. doi: 10.1016/j.mechrescom.2020.103604.
[93]	F. Xu, D. Schillinger, D. Kamensky, V. Varduhn, C. Wang and M. Hsu, The tetrahedral finite cell method for fluids: Immersogeometric analysis of turbulent flow around complex geometries, Computers and Fluids, 141 (2016), 135-154. doi: 10.1016/j.compfluid.2015.08.027.
[94]	S. Xu, B. Gao, M. Hsu and B. Ganapathysubramanian, A residual-based variational multiscale method with weak imposition of boundary conditions for buoyancy-driven flows, Computer Methods in Applied Mechanics and Engineering, 352 (2019), 345-368. doi: 10.1016/j.cma.2019.03.057.
[95]	C. Yang, K. Saurabh, G. Scovazzi, C. Canuto, A. Krishnamurthy and B. Ganapathysubramanian, Optimal surrogate boundary selection and scalability studies for the shifted boundary method on octree meshes, Computer Methods in Applied Mechanics and Engineering, 419 (2024), 116686. doi: 10.1016/j.cma.2023.116686.
[96]	C. Yang, G. Scovazzi, A. Krishnamurthy and B. Ganapathysubramanian, Octree-based shifted boundary method for multiphysics simulations using linearized navier-stokes in complex geometries, preprint, arXiv: 2501.00143, 2024.
[97]	C. Yang, G. Scovazzi, A. Krishnamurthy and B. Ganapathysubramanian, Simulating incompressible flows over complex geometries using the shifted boundary method with incomplete adaptive octree meshes, preprint, arXiv: 2411.00272, 2024.
[98]	X. Yang, X. Zhang, Z. Li and G. He, A smoothing technique for discrete delta functions with application to immersed boundary method in moving boundary simulations, Journal of Computational Physics, 228 (2009), 7821-7836. doi: 10.1016/j.jcp.2009.07.023.
[99]	X. Zeng, G. Stabile, E. N. Karatzas, G. Scovazzi and G. Rozza, Embedded domain reduced basis models for the shallow water hyperbolic equations with the shifted boundary method, Computer Methods in Applied Mechanics and Engineering, 398 (2022), 115143. doi: 10.1016/j.cma.2022.115143.
[100]	L. Zhang, A. Gerstenberger, X. Wang and W. K. Liu, Immersed finite element method, Computer Methods in Applied Mechanics and Engineering, 193 (2004), 2051-2067. doi: 10.1016/j.cma.2003.12.044.
[101]	Q. Zhu, F. Xu, S. Xu, M. Hsu and J. Yan, An immersogeometric formulation for free-surface flows with application to marine engineering problems, Computer Methods in Applied Mechanics and Engineering, 361 (2019), 112748. doi: 10.1016/j.cma.2019.112748.