## Vestibular efficiently discretize highly irregular domains (as

Vestibular system is a microscopic part of the inner ear in charge of the balance function of the human body. Commonly, dizziness is a symptom revealing its failure. Mathematical models have been developed through the years to study parts of the vestibular system 1–5.

It is possible to study the biomechanics of such biological structure with discrete numerical methods. Nowadays, the most popular numerical method is the finite element method (FEM)6, mainly its linear formulation, which assume the triangle and quadrilaterals shapes for 2D analyses and tetrahedron and hexahedron shapes for 3D.

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Presently, using the computed axial tomography (CAT) imaging technique, it is possible to construct realistic and accurate discrete geometrical models.

However, there are some disadvantages in the finite element technique. Being a mesh dependent numerical method, the FEM requires well balanced meshes. For instances, it is computational challenging to efficiently discretize highly irregular domains (as the biological structures) with uniform and high quality meshes. This process represents a high computational cost. Additionally, the mesh refinement requirement in large deformation problems is also computationally heavy.

In the last decades, the computational mechanics community has been developing other discrete numerical techniques, such as meshless methods 7. These methodologies are competitive and alternative advanced discretization techniques capable to obtain efficiently the solution of several fundamental problems 8.

The discretization step is the main difference between meshless methods and FEM: meshless methods discretize the domain using just an unstructured cloud of nodes 7–11 and FEM discretizes the problem domain with a rigid element mesh.

In meshless’ seminal works, surface fitting or the solution of the partial differential equations (PDE) were the main focus 8. Today, these techniques are used to solve a wide-range of linear and non-linear numerical problems 8 10.

Meshless methods comprise several similar advanced discretization techniques 10. Thus, several categorizations are possible for meshless methods 8 10. One of the main labels for meshless methods is the categorization into approximating meshless methods or interpolating meshless methods 10.

Approximating meshless methods construct their shape functions using approximation functions. Their main disadvantage is the lack of the delta Kronecker property, which hinders the imposition of essential and natural boundary conditions 10. Interpolating meshless methods are capable to construct shape functions possessing the Kronecker delta property (as the ones produced with the FEM). In this case, it is possible to impose the essential and natural boundary conditions using the same FEM techniques 10. In the literature it is possible to find research works comparing both meshless approaches 7–11. Generally, approximating meshless methods are capable to deliver smoother and accurate results. However, interpolating meshless methods allow to easily impose essential and natural boundary conditions, easing the computational effort.

In biomechanics meshless methods are particularly attractive. The complex geometry of the bio-structure can be obtained directly from a medical imaging (CAT scan or the MRI images), associating the nodal position with the voxel position.

The most attractive feature of meshless methods it their capability to discretize the problem domain using directly the pixels (or voxels) spatial information from CAT scans or MRI images 10, 12–16. Furthermore, using the grey scale of medical images, meshless methods are capable to identify several biological structures and then attribute to each node the corresponding material properties 10.

The literature shows that meshless methods have clear advantages over other numerical techniques and are a reliable option in biomechanics computational applications 17, mainly using medical imaging techniques (CAT scan and MRI) 18, 19. Moreover, the remeshing efficiency is one of the advantages of meshless methods over FEM, which could be relevant in the structural analysis biological models 20, 21.

More recently, a meshless method – the Smoothed Particle Hydrodynamics (SPH) – was for the first time extended to the computational analysis of the vestibular system, allowing to study the solid/fluid interaction occurring in this complex biological system 22, 23.

In this work, for the first time, the 2D/3D structural analysis of the cupula of the vestibular system is performed. Thus, the free-vibration analysis of the cupula is performed assuming both the FEM and an interpolating meshless method – the Radial Point Interpolation Method  (RPIM) 12, 24.

This manuscript is organized as follows. In section 2 the RPIM formulation is described with detail. Then, in section 3, the system of equations and the corresponding matrix formulation are presented. In section 4, the 2D/3D numerical models of the cupula are presented and in section 5 the obtained results are shown and discussed. The manuscript ends with section 6 in which the main conclusions and final remarks are presented.