This test case focuses on second-order immersed boundary methods applied to the Poisson equation with flower-shaped immersed boundaries.

This test case focuses on second-order immersed boundary methods applied to the Poisson equation with flower-shaped immersed boundaries.

This test case suite focuses on the second-order immersed boundary methods applied to the Poisson equation with Dirichlet boundary condition. The objectives of this test case are:

1. verify the core of the immersed boundary methods.
2. verify the order of spatial convergence of the methods.

For more details, see [1] and [2].

Configurations

Physical domain

The global boundaries are defined by the box:

\begin{align} [-1, +1]^2 &\quad\text{in 2D,}\\ [-1, +1]^3 &\quad\text{in 3D.} \end{align}

The immersed boundary is flower-shaped as represented next figure, and defined by the parametrization:

\begin{align} \phi \rightarrow (0.02\sqrt{5}, 0.02\sqrt(5)^T + (0.5 + 0.2 sin (R\phi))\mathbf{e_r} \quad \forall \theta \in [0, 2\pi] &\quad\text{in 2D,}\\ (x^2+y^2+z^2=r^2) - ((x \pm r)^2 + y^2 + z^2)=r_f^2) - (x^2 + (y \pm r)^2 + z^2=r_f^2) - (x^2 + y^2 + (z\pm r)^2=r_f^2) &\quad\text{in 3D, where r_f=r/2.} \end{align}

The inner domain is outside the shape.

The following figure shows the temperature field and the flower-shaped immersed domain in 2D.

The following figure shows the temperature field mapped on the isosurface of the zero-distance function of the 3D flower-shaped immersed domain.

Solutions

A trigonometric solution is studied. The source term and the boundary conditions are chosen from the following solution:

\begin{align} T(x,y,z) = sin(\pi x) cos(2\times \pi y) &\quad\text{in 2D,}\\ T(x,y,z) = sin(\pi x) cos(2\times \pi y) cos(2\times \pi z) &\quad\text{in 3D.} \end{align}

in the inner domain. The source term is then

\begin{align} S(x,y,z) = -5 \pi^2 sin(\pi x) cos(2 \pi y) &\quad\text{in 2D,}\\ S(x,y,z) = -9 \pi^2 sin(\pi x) cos(2 \pi y) cos(2\times \pi z) &\quad\text{in 3D.} & \end{align}

Boundary conditions

The global boundary conditions are chosen as follows:

Boundary	Condition	Value
left	Dirichlet	\( T = 0 \)
right	Dirichlet	\( T = 0 \)
bottom	Neumann	\( \partial_{\mathrm n}T = 0 \)
top	Neumann	\( \partial_{\mathrm n}T = 0 \)
back	Neumann	\( \partial_{\mathrm n}T = 0 \)
front	Neumann	\( \partial_{\mathrm n}T = 0 \)

The immersed boundary Dirichlet condition are given by the trigonometric solutions.

Grids

Square grids are used.

Test case list

Label
flower_sinus_2D.nts
sphere6_sinus_3D.nts

Runtime parameters

The following common setting are used:

• The temporal and advection terms of the energy equation are deactivated.
• Several IBM methods are used (see [1,2] for details): direct, linear, LGS, LIS, Q, QO, QGS, QGS, QIS, QISS1, QISS1P3, LGSP3.
• Polynomial interpolation of order 2 and 3 are used.
• In 3D, LIS solver library is used (ILUC preconditioner is efficient for this problem).

Results

Second-order accuracy is observed regardless of the IBM used (except the volume penalization one that is only fisrt order).

2D flower-shape

Direct method:

Mesh	Temperature L1 error	order	Temperature L2 error	order	Temperature Linf error	order
40×40	0.010382159768622896	n/a	0.007353166737498385	n/a	0.012243695462294535	n/a
80×80	0.002485721481546915	2.062	0.0017407523517702279	2.079	0.003400911321187694	1.848
160×160	0.0006290534926285257	1.982	0.00044223457340507175	1.977	0.000946715447878943	1.845
320×320	0.00015657175047993082	2.006	0.00010940966845600196	2.015	0.00023385960032318298	2.017

Linear method:

Mesh	Temperature L1 error	order	Temperature L2 error	order	Temperature Linf error	order
40×40	0.011615434935111604	n/a	0.008494135917934185	n/a	0.02413078418766057	n/a
80×80	0.002874372505893045	2.015	0.002077233229680608	2.032	0.0060836226626515	1.988
160×160	0.0007346533976850386	1.968	0.0005346048813906322	1.958	0.0015849716056456353	1.940
320×320	0.00019043751899794397	1.948	0.00014081028524399254	1.925	0.00047737514503620737	1.731

LGS method:

Mesh	Temperature L1 error	order	Temperature L2 error	order	Temperature Linf error	order
40×40	0.010613763459557977	n/a	0.007488289478944289	n/a	0.012283552825729793	n/a
80×80	0.0025192383361042053	2.075	0.0017537796132653819	2.094	0.0030623895247239297	2.004
160×160	0.0006370589060714494	1.983	0.0004453870502506688	1.977	0.0008988397102784873	1.769
320×320	0.00015818074939795825	2.010	0.00010998042129822066	2.018	0.00022143237221317058	2.021

LIS method:

Mesh	Temperature L1 error	order	Temperature L2 error	order	Temperature Linf error	order
40×40	0.010636089706704658	n/a	0.007501577018872526	n/a	0.012278231182752286	n/a
80×80	0.002527479087896519	2.073	0.0017589069409005016	2.093	0.003035316642566954	2.016
160×160	0.0006389435804169586	1.984	0.0004462174889005859	1.979	0.0008496767674746053	1.837
320×320	0.00015878878566372634	2.009	0.00011035299840478839	2.016	0.00022009373441078672	1.949

Q method:

Mesh	Temperature L1 error	order	Temperature L2 error	order	Temperature Linf error	order
40×40	0.009514921357128668	n/a	0.006641876214807318	n/a	0.015200331117321442	n/a
80×80	0.002376016009908403	2.002	0.0016549450237361575	2.005	0.0037107526012221115	2.034
160×160	0.0005961384268197554	1.995	0.0004148758298680987	1.996	0.0010166987670573446	1.868
320×320	0.0001499521927323731	1.991	0.00010443749100635369	1.990	0.00029759893532799175	1.772

QO method:

Mesh	Temperature L1 error	order	Temperature L2 error	order	Temperature Linf error	order
40×40	0.0095102099370316	n/a	0.006637963607814744	n/a	0.015200898342261904	n/a
80×80	0.0023751812054347778	2.001	0.0016542804522660267	2.005	0.003704876875731733	2.037
160×160	0.0005959465817310061	1.995	0.00041471885385714884	1.996	0.001015651875369672	1.867
320×320	0.000149910175631964	1.991	0.00010440415660007945	1.990	0.00029759935097140033	1.771

QGS method:

Mesh	Temperature L1 error	order	Temperature L2 error	order	Temperature Linf error	order
40×40	0.00901871971138663	n/a	0.006213913466743513	n/a	0.008320275098507302	n/a
80×80	0.0022230793077932347	2.020	0.0015370137651529818	2.015	0.002470768140306334	1.752
160×160	0.0005575910236984012	1.995	0.00038473378574649543	1.998	0.0007514637996037798	1.717
320×320	0.00013736713990723874	2.021	9.451585402282266e-05	2.025	0.0001732368669186135	2.117

QGSO method:

Mesh	Temperature L1 error	order	Temperature L2 error	order	Temperature Linf error	order
40×40	0.009014725894602671	n/a	0.006210844392356418	n/a	0.008319900328710661	n/a
80×80	0.002222325562549749	2.020	0.001536497558636247	2.015	0.0024695339797905502	1.752
160×160	0.0005574217051541412	1.995	0.00038461377783611346	1.998	0.0007515810435708703	1.716
320×320	0.00013733091201680088	2.021	9.44915980870854e-05	2.025	0.0001732366211448788	2.117

QIS method:

Mesh	Temperature L1 error	order	Temperature L2 error	order	Temperature Linf error	order
40×40	0.008868231198472656	n/a	0.006107703517977893	n/a	0.008331126149572432	n/a
80×80	0.0021797976063784655	2.024	0.001510935780416645	2.015	0.0020166518315175574	2.047
160×160	0.0005450854684979009	2.000	0.00037636499468866885	2.005	0.0005605451713657228	1.847
320×320	0.0001339208124446688	2.025	9.242005827090916e-05	2.026	0.0001608484462453097	1.801

QISS1 method:

Mesh	Temperature L1 error	order	Temperature L2 error	order	Temperature Linf error	order
40×40	0.010382159768623025	n/a	0.007353166737498444	n/a	0.012243695462294313	n/a
80×80	0.0024857214815461673	2.062	0.0017407523517697888	2.079	0.003400911321187583	1.848
160×160	0.0006290415905712212	1.982	0.00044223287668653595	1.977	0.0009467154415244705	1.845
320×320	0.0001565717504854494	2.006	0.00010940966845948416	2.015	0.00023385960032318298	2.017

QISS1P3 method:

Mesh	Temperature L1 error	order	Temperature L2 error	order	Temperature Linf error	order
40×40	0.007041580518529886	n/a	0.005342061379365422	n/a	0.00899006179897821	n/a
80×80	0.0017842081972230008	1.981	0.0013498408606690679	1.985	0.0022855124003658966	1.976
160×160	0.00044942063410977214	1.989	0.0003390984474031629	1.993	0.0013408613899511534	0.769
320×320	0.00011285013360274934	1.994	8.49320413804135e-05	1.997	0.00014335355381078774	3.226
640×640	2.828965321957408e-05	1.996	2.1264977081482812e-05	1.998	3.5864566322318936e-05	1.999

LGSP3 method:

Mesh	Temperature L1 error	order	Temperature L2 error	order	Temperature Linf error	order
40×40	0.009519487523919427	n/a	0.0067197085370243265	n/a	0.01851760976561312	n/a
80×80	0.0023841131724082042	1.997	0.0016653593189984797	2.013	0.004530134493496707	2.031
160×160	0.0006130742115801825	1.959	0.0004303497526787066	1.952	0.0011960010010085842	1.921
320×320	0.00015892894996284544	1.948	0.00011382373798071934	1.919	0.0003918649150661535	1.610
640×640	3.8554524015091366e-05	2.043	2.6994695199321087e-05	2.076	0.0001321051454794553	1.569

VOLUME PENALIZATION method:

Mesh	Temperature L1 error	order	Temperature L2 error	order	Temperature Linf error	order
20×20	0.1423517275256972	n/a	0.13457673412764698	n/a	0.3463818132133143	n/a
40×40	0.05086705036850881	1.485	0.04365649591792538	1.624	0.11864388079782373	1.546
80×80	0.022268013554225887	1.192	0.019496305658366717	1.163	0.06774546049852681	0.808
160×160	0.012123104585662885	0.877	0.010269191309689703	0.925	0.03678209779122199	0.881

3D flower-shape

Direct method:

Mesh	Temperature L1 error	order	Temperature L2 error	order	Temperature Linf error	order
16×16×16	0.002530207803768791	n/a	0.003733806511973472	n/a	0.014717502590773535	n/a
32×32×32	0.0006572926338986642	1.945	0.0009643207556546832	1.953	0.005709020961563116	1.366
64×64×64	0.00016561935601120702	1.989	0.00024165628655621659	1.997	0.0015339800512155222	1.896
128×128×128	4.1708089561245926e-05	1.989	6.089729974985724e-05	1.989	0.00038998327954020695	1.976

Linear method:

Mesh	Temperature L1 error	order	Temperature L2 error	order	Temperature Linf error	order
16×16×16	0.002998151122443975	n/a	0.004344507049945625	n/a	0.015372675355382559	n/a
32×32×32	0.0007760167929225879	1.950	0.0011032211607128663	1.977	0.006081335961378609	1.338
64×64×64	0.00021220469451200852	1.871	0.00029748650250597516	1.891	0.0015224266925817041	1.998
128×128×128	5.238225738317134e-05	2.018	7.34889507721573e-05	2.017	0.00037937521947750685	2.005

LGS method:

Mesh	Temperature L1 error	order	Temperature L2 error	order	Temperature Linf error	order
16×16×16	0.0030618960018274867	n/a	0.004426410858609621	n/a	0.01537267474292936	n/a
32×32×32	0.0007760442297105378	1.980	0.0010917127018378403	2.020	0.00536361121548512	1.519
64×64×64	0.00021048441166218152	1.882	0.0002934510516660949	1.895	0.0015220920554638515	1.817
128×128×128	5.2515142637114835e-05	2.003	7.32361190094301e-05	2.002	0.000375996389903932	2.017

LIS method:

Mesh	Temperature L1 error	order	Temperature L2 error	order	Temperature Linf error	order
16×16×16	0.0030618960018274867	n/a	0.004426410858609621	n/a	0.01537267474292936	n/a
32×32×32	0.0007830927238077406	1.967	0.001100594886892528	2.008	0.005357472218672887	1.521
64×64×64	0.00021314995973990865	1.877	0.00029698259457119094	1.890	0.0015222424285000029	1.815
128×128×128	5.293386831882026e-05	2.010	7.376969235673956e-05	2.009	0.0003719320602803	2.033

Q method:

Mesh	Temperature L1 error	order	Temperature L2 error	order	Temperature Linf error	order
16×16×16	0.002544121596646575	n/a	0.0037373296876410216	n/a	0.011561598235916848	n/a
32×32×32	0.0006661378964774928	1.933	0.0009755616996504988	1.938	0.004223717992182352	1.453
64×64×64	0.00016619058552379788	2.003	0.00024225240871216888	2.010	0.0010069523021716265	2.069
128×128×128	4.1520792301227795e-05	2.001	6.054194327995673e-05	2.001	0.000307075726406536	1.713

QO method:

Mesh	Temperature L1 error	order	Temperature L2 error	order	Temperature Linf error	order
16×16×16	0.0025445500423156454	n/a	0.003737823472207631	n/a	0.011554241366261708	n/a
32×32×32	0.0006659191042271541	1.934	0.0009753089238174323	1.938	0.004223925903712233	1.452
64×64×64	0.00016615818042739684	2.003	0.00024221558973953192	2.010	0.0010067035451161166	2.069
128×128×128	4.151662956003155e-05	2.001	6.053677413596293e-05	2.000	0.0003070798818106679	1.713

QGS method:

Mesh	Temperature L1 error	order	Temperature L2 error	order	Temperature Linf error	order
16×16×16	0.002544121596646575	n/a	0.0037373296876410216	n/a	0.011561598235916848	n/a
32×32×32	0.0006647141739076339	1.936	0.0009702299262975187	1.946	0.003962371841081724	1.545
64×64×64	0.0001660843181364729	2.001	0.00024212344755499376	2.003	0.0010352210820258057	1.936
128×128×128	4.140589733191384e-05	2.004	6.037272080539582e-05	2.004	0.00031957223105405497	1.696

QGSO method:

Mesh	Temperature L1 error	order	Temperature L2 error	order	Temperature Linf error	order
16×16×16	0.0025445500423156454	n/a	0.003737823472207631	n/a	0.011554241366261708	n/a
32×32×32	0.0006644957069418157	1.937	0.0009699777136939204	1.946	0.003962421039037767	1.544
64×64×64	0.0001660524268214208	2.001	0.00024208726915003547	2.002	0.0010350363354507675	1.937
128×128×128	4.140186734511885e-05	2.004	6.036786865976097e-05	2.004	0.00031957791968051374	1.695

QIS method:

Mesh	Temperature L1 error	order	Temperature L2 error	order	Temperature Linf error	order
16×16×16	0.002544121596646575	n/a	0.0037373296876410216	n/a	0.011561598235916848	n/a
32×32×32	0.0006637656779331995	1.938	0.0009685862225868512	1.948	0.0038100624914474412	1.601
64×64×64	0.00016576666976389278	2.002	0.00024170269431509085	2.003	0.0009816815477917684	1.956
128×128×128	4.126214650815817e-05	2.006	6.018138707808345e-05	2.006	0.000301636729495236	1.702

QISS1 method:

Mesh	Temperature L1 error	order	Temperature L2 error	order	Temperature Linf error	order
16×16×16	0.0025961584611331003	n/a	0.003829932361667098	n/a	0.01471672541694602	n/a
32×32×32	0.00067497029512167	1.943	0.000989766745164492	1.952	0.0056133313436673715	1.391
64×64×64	0.000170351580835294	1.986	0.0002481066081926968	1.996	0.00151163675444288	1.893
128×128×128	4.31595494354186e-05	1.981	6.286766855923313e-05	1.981	0.00038589492926466296	1.970

VOLUME PENALIZATION method:

Mesh	Temperature L1 error	order	Temperature L2 error	order	Temperature Linf error	order
16×16×16	0.012364887047713448	n/a	0.016465244524214943	n/a	0.06632454724188042	n/a
32×32×32	0.008934858153927734	0.469	0.011364784275953985	0.535	0.07136614432590843	-0.106
64×64×64	0.005111332788139465	0.806	0.006299093573720937	0.851	0.02839942993900557	1.329
128×128×128	0.002586323353924102	0.983	0.0032173568057359448	0.969	0.018854617226945425	0.591

References

[1] J. Picot, S. Glockner, Discretization stencil reduction of direct forcing immersed boundary methods on rectangular cells: the Ghost Node Shifting Method, Journal of Computational Physics, 364, pp18-48, 2018.

[2] A. M. D. Jost and S. Glockner, Direct forcing immersed boundary methods: Improvements to the Ghost Node Method, Journal of Computational Physics, volume 438, 110371, 2021.