Vascularized tissues and fiber reinforced materials (FRMs) can be modeled as bi-phasic materials, where different constitutive behaviors are associated with different phases. 

The numerical study of these problems through a full geometrical resolution of the two phases is often computationally infeasible: the usual approach to model these problems is to use homogenization theory, and exploit strong regularity assumptions on the distribution of the one-dimensional topology. 

Both approaches fall short in intermediate regimes where lack of regularity does not justify a homogenized approach, and when the complexity of the one-dimensional network renders the fully resolved problem numerically intractable.

Our proposal is to use distributed Lagrange multipliers where the two phases are coupled through a constraint condition on non-matching, overlapping geometries, opening the way for intricate fiber-bulk and vasculature-bulk couplings as well as allowing complex geometries with no alignment requirements between the discretization of the background problem and the one-dimensional topology.

• L. Heltai and P. Zunino, Reduced lagrange multiplier approach for non-matching coupling of mixed- dimensional domains, Mathematical Models and Methods in Applied Sciences 33 (2023), no. 12, 2425– 2462.

•  G. Alzetta and L. Heltai. Multiscale modeling of fiber reinforced materials via non-matching immersed methods. Computers & Structures, 239:106334, Oct. 2020.

• L. Heltai and A. Caiazzo. Multiscale modeling of vascularized tissues via non-matching immersed methods. International Journal for Numerical Methods in Biomedical Engineering, 35(12):e3264, 2019.

• L. Heltai, A. Caiazzo, and L. Müller. Multiscale coupling of one-dimensional vascular models and elastic tissues. Annals of Biomedical Engineering, 2021.

• J. Joachim, C. A. Daunais, V. Bibeau, L. Heltai, and B. Blais, A parallel and adaptative nitsche immersed boundary method to simulate viscous mixing, Journal of Computational Physics (2023), 112189.

• L. Heltai and W. Lei. A priori error estimates of regularized elliptic problems. Numerische Mathematik, 146(3):571–596, 2020.

• L. Heltai and N. Rotundo. Error estimates in weighted sobolev norms for finite element immersed interface methods. Computers & Mathematics with Applications, 78(11):3586–3604, Dec. 2019.

When solving elliptic partial differential equations in a region containing immersed interfaces (possibly evolving in time), it is often desirable to approximate the problem using an independent background discretization, not aligned with the interface itself. In non-matching methods, the interface is replaced by a singular force field that produces the desired interface conditions, as done in immersed boundary methods.

Two different approaches are used in the literature for the approximation of such problems:

• variational based singular forcing terms;

• regularized singular forging term.

In the first class of methods, classical finite element approximations are known to have inferior convergence properties, when compared to matching methods. We show, however, that the detrimental effect due to the non-matching nature of the algorithm is only a local phenomenon, restricted to a small neighborhood of the interface. Optimal approximations can be constructed in a natural and inexpensive way, by reformulating the problem in weighted Sobolev spaces.

Smoothed Adaptive Finite Elements (S-AFEM) and Smoothed Adaptive Perturbed Inverse Iteration (SA-PINVIT) are innovative Finite Element algorithms for the solution of source and eigenvalue problems with local adaptivity. 

The algorithms are inspired by the ascending phase of the V-cycle multigrid method: we replace accurate algebraic solutions in intermediate cycles of the classical AFEM and PINVIT methods with the application of a prolongation step, followed by a smoother. 

Even though these intermediate solutions are far from the exact algebraic solutions, their a-posteriori error estimation produces a refinement pattern that is substantially equivalent to the one that would be generated by classical AFEM, at a considerable fraction of the computational cost.

Intermediate solution steps are instrumental for the construction of the finally adapted grid, and play no role in the final solution, which is the only one that is retained for analysis and processing.

S-AFEM and SA-PINVIT are simple yet effective algorithms to reduce the overall computational cost of AFEM and PINVIT, by providing a fast procedure for the construction of quasi-optimal mesh sequences that do not require the exact solution of the algebraic problem in the intermediate loops.

• S. Giani, L. Grubisic, L. Heltai, and O. Mulita. Smoothed-adaptive perturbed inverse iteration for elliptic eigenvalue problems. Computational Methods in Applied Mathematics, 21(2):385–405, 2021.

• O. Mulita, S. Giani, and L. Heltai. Quasi-optimal mesh sequence construction through smoothed adaptive finite element methods. SIAM Journal on Scientific Computing, 2021. To appear.

• L. Heltai, J. Kiendl, A. DeSimone, and A. Reali. A natural framework for isogeometric fluid-structure in- teraction based on bem-shell coupling. Computer methods in applied mechanics and engineering, 316:522– 546, 04/2017 2017.

• L. Heltai, M. Arroyo, and A. DeSimone. Nonsingular isogeometric boundary element method for stokes flows in 3d. Computer Methods in Applied Mechanics and Engineering, 268:514–539, 2014.

Iso-geometric analysis (IGA) bypasses mesh generation and allows one to perform direct design-to-analysis simulations, by employing the same class of functions used for geometry parameterization in CAGD packages during the analysis process.

Most modern CAD tools, however, are based on boundary representation (B-Rep) objects, making the use of volume-based finite element isogeometric analysis tools (FE-IGA) less attractive, since they require the extension of the computational domain inside (or outside) the enclosing (or enclosed) CAGD surface.

For thin structures, isogeometric shell models circumvent this issue since they only need a surface description of the structure. 

For fluid dynamics, isogeometric boundary element methods (IGA-BEM) also circumvent the issue mentioned above by reformulating the volumetric flow problem in boundary integral form. 

Such dimensionality reduction makes the coupling between boundary integral formulations and shell theory an ideal combination for a large class of fluid–structure interaction (FSI) problems, where thin structures interact with Newtonian incompressible flows, and it fits ideally in the IGA paradigm, by only requiring surface representations for both fluid and structural analyses.

Immersed methods find their most famous application in the solution of general fluid–structure interaction (FSI) problems.

In immersed methods the solid domain is surrounded by the fluid. When the fluid and solid do not slip relative to one another, these methods have three basic features:

1. The support of the equations of motion of the fluid is extended to the union of the physical fluid and solid domains.

2. The equations of motion of the fluid have terms that, from a continuum mechanics viewpoint, are body forces ‘‘informing’’ the fluid of its interaction with the solid.

3. The velocity field of the immersed solid is identified with the restriction to the solid domain of the velocity field in the equations of motion of the fluid.

Efficient numerical implementation of such methods is the subject of many research and projects.  A toy example code is included in the tutorial step-70 of the deal.II library.

• D. Boffi, L. Gastaldi, and L. Heltai. A Distributed Lagrange Formulation of the Finite Element Immersed Boundary Method for Fluids Interacting with Compressible Solids, pages 1–21. Springer International Publishing, Cham, 2018.

• S. Roy, L. Heltai, and F. Costanzo. Benchmarking the immersed finite element method for fluid-structure interaction problems. Computers and Mathematics with Applications, 69:1167–1188, 2015.

• L. Heltai and F. Costanzo. Variational implementation of immersed finite element methods. Computer Methods in Applied Mechanics and Engineering, 229–232(0):110 – 127, 2012.