The Virtual Element Method with curved edges

In this paper we initiate the investigation of Virtual Elements with curved faces. We consider the case of a fixed curved boundary in two dimensions, as it happens in the approximation of problems posed on a curved domain or with a curved interface. While an approximation of the domain with polygons leads, for degree of accuracy $k \geq 2$, to a sub-optimal rate of convergence, we show (both theoretically and numerically) that the proposed curved VEM lead to an optimal rate of convergence

Basic principles of hp Virtual Elements on quasiuniform meshes

In the present paper we initiate the study of $hp$ Virtual Elements. We focus on the case with uniform polynomial degree across the mesh and derive theoretical convergence estimates that are explicit both in the mesh size $h$ and in the polynomial degree $p$ in the case of finite Sobolev regularity. Exponential convergence is proved in the case of analytic solutions. The theoretical convergence results are validated in numerical experiments. Finally, an initial study on the possible choice of local basis functions is included

Serendipity Face and Edge VEM Spaces

We extend the basic idea of Serendipity Virtual Elements from the previous case (by the same authors) of nodal ($H^1$-conforming) elements, to a more general framework. Then we apply the general strategy to the case of $H(div)$ and $H(curl)$ conforming Virtual Element Methods, in two and three dimensions

Serendipity Nodal VEM spaces

We introduce a new variant of Nodal Virtual Element spaces that mimics the "Serendipity Finite Element Methods" (whose most popular example is the 8-node quadrilateral) and allows to reduce (often in a significant way) the number of internal degrees of freedom. When applied to the faces of a three-dimensional decomposition, this allows a reduction in the number of face degrees of freedom: an improvement that cannot be achieved by a simple static condensation. On triangular and tetrahedral decompositions the new elements (contrary to the original VEMs) reduce exactly to the classical Lagrange FEM. On quadrilaterals and hexahedra the new elements are quite similar (and have the same amount of degrees of freedom) to the Serendipity Finite Elements, but are much more robust with respect to element distortions. On more general polytopes the Serendipity VEMs are the natural (and simple) generalization of the simplicial case

Lowest order Virtual Element approximation of magnetostatic problems

We give here a simplified presentation of the lowest order Serendipity Virtual Element method, and show its use for the numerical solution of linear magneto-static problems in three dimensions. The method can be applied to very general decompositions of the computational domain (as is natural for Virtual Element Methods) and uses as unknowns the (constant) tangential component of the magnetic field $\mathbf{H}$ on each edge, and the vertex values of the Lagrange multiplier $p$ (used to enforce the solenoidality of the magnetic induction $\mathbf{B}=\mu\mathbf{H}$). In this respect the method can be seen as the natural generalization of the lowest order Edge Finite Element Method (the so-called "first kind N\'ed\'elec" elements) to polyhedra of almost arbitrary shape, and as we show on some numerical examples it exhibits very good accuracy (for being a lowest order element) and excellent robustness with respect to distortions

Recognizing Activities of Daily Living of People with Parkinson's

Tese de mestrado, Informática, Universidade de Lisboa, Faculdade de Ciências, 2022Parkinson's disease is a common neurodegenerative disease that affects a large part of the world's population. This disease involves a lot of symptoms, however the most prevalent is the change in the patient's movements or even the loss of functionality. There is no treatment, however it exists medication that relieves and reduces the symptoms for a period. A Parkinson’s patient needs to be watched by clinicians to understand if the medication is working correctly and to analyse the disease progression. The current way of doing this evaluation is at clinics where the patient needs to go to the clinic or to live there. With this into consideration it was requested a monitoring system of activities of daily living for Parkinson’s patient. The monitoring system consists in a mobile application in an Android smartphone serving as a diary for the patient of clinician to record the activities done at that moment. With this application, the patient needs to wear an accelerometer in the wrist to gather the acceleration in the 3-axis. The application besides the monitoring function, it gives the ability to the clinician to schedule lists of activities for the patient to do during the day, allowing the clinician to have some control. We carried out a study with 10 healthy participants which used the monitorization system for 3 days each. The patient would worn the accelerometer and record the activities that they would do throughout the day, was asked a minimum of 5 activities per day. Alongside this recording it was schedule 1 list of activities to be carried out each day, this list only had motor activities such as walk, sit down, and stand up. At the end of each participant study, it was made a questionnaire with standard usability questions and an interview that helped us understand if the system was reliable or not