Interpolation for nonlinear BVP in circular membrane with known upper and lower solutions

AbstractA successive nonextrapolatory linear interpolation is described to solve a singular two-point boundary value problem arising in circular membrane theory. The problem is associated with a second-order nonlinear ordinary differential equation for which the upper and lower bounds of the solution is analytically established/known. The importance and the scope of these bounds in solving the problem is stressed. Also depicted graphically are the lower and upper solutions as well as the true and iterated solutions. In addition, discussed are the reasons why linear interpolation, and not nonlinear interpolation or bisection which are possible procedures, has been employed

Wave impact in sloshing flows: Hydroelasticity in shallow water condition

The hydroelastic interaction between a fluid (sloshing flow) and a flexible metal structure, as a consequence of a gravitational wave impact, has been investigated. When hydroelasticity occurs, the real stresses which the structure must overcome, may be underestimate if only the hydrodynamic pressure is taken in account. The knowledge of the stresses acting on the structure, as well as, the physical mechanisms which are able to trigger this kind of phenomena, are fundamental for a right design and the safety of a marine structures. The investigation has been performed following both the experimental and mathematical approach. An experimental set-up has been designed for the reproduction of the impact, during a two dimensional sloshing flow in low filling condition, against a flexible structure(, as well as, a rigid one). Two specific typologies of wave impacts have been considered, both of them, characterized by hydrodynamic loads which may activate hydroelastic effects on the structure: a) Flip-Through type wave impact b) Single Air-Bubble entrapment wave impact for the last one, the investigation has been extended also at the influence of the Euler number on the structural stress. The influence of the ullage pressure is an important topic related to the scaling procedure when model scale experiments are performed. For a better identification of the major physical features which play an active role in the hydroelastic phenomena, an hybrid “weak” hydroelastic and a fully hydroelastic methods have been developed. The three different sub-problems, that have been individuated from the experimental activities: 1a) sloshing stage with single phase flow, 1b) sloshing stage with two phase flow and 2) structural problem, have been modelled with a proper mathematical models, where the physical assumptions adopted, have been inspired by the experimental finding and also by the literature. The hybrid model combines a numerical model for the structural problem with hydrodynamic loads estimated during experimental tests with a fully rigid structure. More in detail, the Euler beam theory together with a model for the added mass have been used to describe the behavior of the structure. The hybrid model highlights how the added mass effect influence both the natural frequencies and the displacement of the structure. Anyway some differences on the structure displacement have been observed, especially on the higher peaks just after the impact, this suggests that a stronger hydroelastic interaction is present. For the fully hydroelastic model, also the sloshing sub-problem, which can be considered both as single phase of two phase flow, depending on the impact type, has been solved numerically. In particular, a mixed Eulerian-Lagrangian method has been applied for the evolution of the free surface, where for the liquid phase the hypotheses of incompressible and irrotational fluid have been considered. In that cases where the air cavity is present, the pressure inside the cavity has been modelled with ad-hoc semi-analytical model, such as the “lumped” model. The dynamic behaviour of the structure has been approximated as in the hybrid method. The coupling, during the numerical time integration scheme, of the cited sub-models gives the fully hydroelastic method

Fast, High-order Algorithms for Simulating Vesicle Flows Through Periodic Geometries.

This dissertation presents a new boundary integral equation (BIE) method for simulating vesicle flows through periodic geometries. We begin by describing the periodization scheme, in the absence of vesicles, for singly and doubly periodic geometries in 2 dimensions and triply periodic geometries in three dimensions. Later, the periodization scheme will be expanded to include multiple vesicles confined by singly periodic channels of arbitrary shape. Rather than relying on the periodic Green’s function as classical BIE methods do, the method combines the free-space Green’s function with a small auxiliary basis and imposes periodicity as an extra linear condition. As a result, we can exploit existing free-space solver libraries, quadratures, and fast algorithms to handle a large number of vesicles in a geometrically complex domain. Spectral accuracy in space is achieved using the periodic trapezoid rule and product quadratures, while a first-order semi-implicit scheme evolves particles by treating the vesicle-channel interactions explicitly. New constraint-correction formulas are introduced that preserve reduced areas of vesicles, independent of the number of time steps taken. By using two types of fast algorithms, (i) the fast multipole method (FMM) for the computation of the vesicle-vesicle and the vesicle-channel hydrodynamic interaction, and (ii) a fast direct solver for the BIE on the fixed channel geometry, the computational cost is reduced to O(N) per time step where N is the spatial discretization size. We include two example applications that utilize BIE methods with periodic boundary conditions. The first seeks to determine the equilibrium shapes of periodic planar elastic membranes. The second models the opening and closing of mechanosensitive (MS) channels on the membrane of a vesicle when exposed to shear stress while passing through a constricting channel.PHDApplied and Interdisciplinary MathematicsUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/135778/1/gmarple_1.pd

A surrogate model for computational homogenization of elastostatics at finite strain using high-dimensional model representation-based neural network

We propose a surrogate model for two-scale computational homogenization of elastostatics at finite strains. The macroscopic constitutive law is made numerically available via an explicit formulation of the associated macroenergy density. This energy density is constructed by using a neural network architecture that mimics a high-dimensional model representation. The database for training this network is assembled through solving a set of microscopic boundary value problems with the prescribed macroscopic deformation gradients (input data) and subsequently retrieving the corresponding averaged energies (output data). Therefore, the two-scale computational procedure for nonlinear elasticity can be broken down into two solvers for microscopic and macroscopic equilibrium equations that work separately in two stages, called the offline and online stages. The finite element method is employed to solve the equilibrium equation at the macroscale. As for microscopic problems, an FFT-based collocation method is applied in tandem with the Newton-Raphson iteration and the conjugate-gradient method. Particularly, we solve the microscopic equilibrium equation in the Lippmann-Schwinger form without resorting to the reference medium. In this manner, the fixed-point iteration that might require quite strict numerical stability conditions in the nonlinear regime is avoided. In addition, we derive the projection operator used in the FFT-based method for homogenization of elasticity at finite strain

Novel Discretization Schemes for the Numerical Simulation of Membrane Dynamics

Motivated by the demands of simulating flapping wings of Micro Air Vehicles, novel numerical methods were developed and evaluated for the dynamic simulation of membranes. For linear membranes, a mixed-form time-continuous Galerkin method was employed using trilinear space-time elements, and the entire space-time domain was discretized and solved simultaneously. For geometrically nonlinear membranes, the model incorporated two new schemes that were independently developed and evaluated. Time marching was performed using quintic Hermite polynomials uniquely determined by end-point jerk constraints. The single-step, implicit scheme was significantly more accurate than the most common Newmark schemes. For a simple harmonic oscillator, the scheme was found to be symplectic, frequency-preserving, and conditionally stable. Time step size was limited by accuracy requirements rather than stability. The spatial discretization scheme employed a staggered grid, grouping of nonlinear terms, and polygon shape functions in a strong-form point collocation formulation. Validation against existing experimental data showed the method to be accurate until hyperelastic effects dominate

A surrogate model for computational homogenization of elastostatics at finite strain using high‐dimensional model representation‐based neural network

We propose a surrogate model for two-scale computational homogenization of elastostatics at finite strains. The macroscopic constitutive law is made numerically available via an explicit formulation of the associated macroenergy density. This energy density is constructed by using a neural network architecture that mimics a high-dimensional model representation. The database for training this network is assembled through solving a set of microscopic boundary value problems with the prescribed macroscopic deformation gradients (input data) and subsequently retrieving the corresponding averaged energies (output data). Therefore, the two-scale computational procedure for nonlinear elasticity can be broken down into two solvers for microscopic and macroscopic equilibrium equations that work separately in two stages, called the offline and online stages. The finite element method is employed to solve the equilibrium equation at the macroscale. As for microscopic problems, an FFT-based collocation method is applied in tandem with the Newton-Raphson iteration and the conjugate-gradient method. Particularly, we solve the microscopic equilibrium equation in the Lippmann-Schwinger form without resorting to the reference medium. In this manner, the fixed-point iteration that might require quite strict numerical stability conditions in the nonlinear regime is avoided. In addition, we derive the projection operator used in the FFT-based method for homogenization of elasticity at finite strain

MATLAB

A well-known statement says that the PID controller is the "bread and butter" of the control engineer. This is indeed true, from a scientific standpoint. However, nowadays, in the era of computer science, when the paper and pencil have been replaced by the keyboard and the display of computers, one may equally say that MATLAB is the "bread" in the above statement. MATLAB has became a de facto tool for the modern system engineer. This book is written for both engineering students, as well as for practicing engineers. The wide range of applications in which MATLAB is the working framework, shows that it is a powerful, comprehensive and easy-to-use environment for performing technical computations. The book includes various excellent applications in which MATLAB is employed: from pure algebraic computations to data acquisition in real-life experiments, from control strategies to image processing algorithms, from graphical user interface design for educational purposes to Simulink embedded systems

Formulation and implementation of conforming finite element approximations to static and eigenvalue problems for thin elastic shells

Bibliography: pages 132-135.In deriving asymptotic error estimates for a conforming finite element analyses of static thin elastic shell problems, the French mathematician Ciarlet (1976) proposed an approach to the formulation of such problems. The formulation he uses is based on classical shell theory making use of Kirchhoff-Koiter assumptions. The shell problem is posed in two-dimensional space to which the real problem, in three-dimensional space, is related by a mapping of the domain of the problem to the shell mid-surface. The finite element approximation is formulated in terms of the covariant components of the shell mid-surface displacement field. In this study, Ciarlet's formulation is extended to include the eigenvalue problem for the shell. In addition to this, the aim of the study is to obtain some indication of how well this approach might be expected to work in practice. The conforming finite element approximation of both the static and eigenvalue problems are implemented. Particular attention is paid to allowing generality of the shell surface geometry through the use of an approximate mapping. The use of different integration rules, in-plane displacement component interpolation schemes and approximate geometry schemes are investigated. Results are presented for shells of different geometries for both static and eigenvalue analyses; these are compared with independently obtained results