A GPU Accelerated Immersed Boundary Method for Linear Elasticity

     Immersed Boundary Methods (IBMs) have been extensively used for fluid-structure interaction (FSI) and fluid dynamics simulations. IBMs can also be implemented to solve linear elasticity problems. A SIBM is already developed by yours truly in abundant spare time, of course 😼. The developed SIBM is  now adapted to solve the linear-elasticity problems 〰️.

     SIBM offers several advantages compared to legacy finite element methods. With SIBM, there is no need to create expensive 🤑 and time-consuming ⏱︎  meshes for deforming boundaries. SIBM embeds objects of any arbitrary shapes on a simple Cartesian grid. The SIBM works well for domains with irregular or those boundaries which are evolving with time. In such cases legacy meshes become impractical.

     The ray casting algorithm🛸; a fundamental 🧱 technique used in video game 🎮 development and computer graphics, has been implemented in a similar manner to the fluid dynamics version. The detailed description of the method is already presented here.

     NOTE: As is the case with the fluid dynamics version, this code also requires a GPU to run. If dear readers don't have a GPU; a solution is to please stop being peasants... 🙉

Brazilian Test

     The first case presented is the case of indirect tensile test also referred to as the Brazilian test. The boundary conditions are described in Fig. 1. The simple boundary conditions include a load at the top in compression with a vertical support at the bottom. Due to the geometry of the specimen, the stress field center portion comes under tension. Meanwhile, the results from the simulations are presented in Fig. 2. It is clear that the SIBM can handle curved boundary problems with relative ease on a simple cartesian grid, without complicated the meshing.

Fig. 1, The boundary conditions

Fig. 2, The simulation results

Plate in Tension with a Hole

     This famous 📰 benchmark case of linear-elasticity 〰️ is now solvable with high accuracy. The challenge in this simple case is about application of the boundary conditions on the curved portion i.e. the hole. This is where IBM really shines. As the name suggests, the IBM immerses the hole inside grid. This makes application of the boundary 📏 conditions simple and straight forward. The boundary conditions are shown in Fig. 3. In the SIBM code, full plate is simulated. Note that this case has already been validated using a meshless 🕸️ method by yours truly 🤓. More details 📚 and free 💸 code  is available here.

Fig. 3, The boundary conditions

     The validation of this particular case has been extensively published. A not to be named 🤫, pre-validated commercial 🤑 code is used to validate the SIBM. The result from the commercial solver predict the von-Mises stress to be at 3.39 MPa. The SIBM indicate the von-Mises stress to be at, 🥁, 3.49 MPa ❗The resultant displacement from the 💰 code are at 0.000554 mm as compared to the SIBM value of 0.0006 mm. The resulting equivalent strain from the commercial code is at 1.31e-5. The strain from the SIBM is at 1.78e-5. The results from SIBM are plotted in Fig. 4. Within Fig. 4, the deformed plate is shown using red color while the undeformed plate is shown using black color. Please note that in the actual simulation, there are no moving nodes. This is another important advantage of SIBM which results in a much simple algorithm.

     The von-Mises stress, equivalent strain and resultant displacement (bottom) are shown in Fig. 5. The results shown in Fig. 5 are on an undeformed plate, for clarity. Within Fig. 5, the color red 🩷 means maximum and the color blue 💙 means minimum value. It can be seen 👓 that the code captures stress concentrations 🧐 with high accuracy.

Fig. 4, Deformed VS undeformed plate

Fig. 5, The results from post-processing

NAFEMS Elliptical Membrane

     The National Agency for Finite Element Methods and Standards is a long name 😫. Anyways 🙈, this case is the benchmark for testing irregular and curved 〜 boundaries domains, mixed 🥘 boundary conditions and symmetry 🪞. The SIBM  avoids time-consuming and complex meshes to capture curved 〰️ boundaries by embedding ☂️ the geometry in a regular cartesian grid. The result is accurate simulation of complex cases such as the symmetric elliptical ⬭ membrane. The peak stress of 98.05 MPa is predicted from the SIBM as compared to 92.7 MPa from the published literature, on a very coarse mesh 💠. The boundary conditions are shown in Fig. 6. The predicted normal stress σᵧᵧ is shown in Fig. 7. The deformed membrane (red) is also shown within Fig.7.

Fig. 6, The boundary conditions

Fig. 7, The results

     Thank you for reading! If you want to hire me as a post-doc researcher in the fields of thermo-fluids and / or fracture mechanics, do reach out!