Meshless methods for shear-deformable beams and plates based on mixed weak forms

Thin structural theories such as the shear-deformable Timoshenko beam and Reissner-Mindlin plate theories have seen wide use throughout engineering practice to simulate the response of structures with planar dimensions far larger than their thickness dimension. Meshless methods have been applied to construct numerical methods to solve the shear deformable theories. Similarly to the finite element method, meshless methods must be carefully designed to overcome the well-known shear-locking problem. Many successful treatments of shear-locking in the finite element literature are constructed through the application of a mixed weak form. In the mixed weak form the shear stresses are treated as an independent variational quantity in addition to the usual displacement variables. We introduce a novel hybrid meshless-finite element formulation for the Timoshenko beam problem that converges to the stable first-order/zero-order finite element method in the local limit when using maximum entropy meshless basis functions. The resulting formulation is free from the effects shear-locking. We then consider the Reissner-Mindlin plate problem. The shear stresses can be identified as a vector field belonging to the Sobelov space with square integrable rotation, suggesting the use of rotated Raviart-Thomas-Nedelec elements of lowest-order for discretising the shear stress field. This novel formulation is again free from the effects of shear-locking. Finally we consider the construction of a generalised displacement method where the shear stresses are eliminated prior to the solution of the final linear system of equations. We implement an existing technique in the literature for the Stokes problem called the nodal volume averaging technique. To ensure stability we split the shear energy between a part calculated using the displacement variables and the mixed variables resulting in a stabilised weak form. The method then satisfies the stability conditions resulting in a formulation that is free from the effects of shear-locking.Open Acces

Use of Machine Learning for Automated Convergence of Numerical Iterative Schemes

Convergence of a numerical solution scheme occurs when a sequence of increasingly refined iterative solutions approaches a value consistent with the modeled phenomenon. Approximations using iterative schemes need to satisfy convergence criteria, such as reaching a specific error tolerance or number of iterations. The schemes often bypass the criteria or prematurely converge because of oscillations that may be inherent to the solution. Using a Support Vector Machines (SVM) machine learning approach, an algorithm is designed to use the source data to train a model to predict convergence in the solution process and stop unnecessary iterations. The discretization of the Navier Stokes (NS) equations for a transient local hemodynamics case requires determining a pressure correction term from a Poisson-like equation at every time-step. The pressure correction solution must fully converge to avoid introducing a mass imbalance. Considering time, frequency, and time-frequency domain features of its residual’s behavior, the algorithm trains an SVM model to predict the convergence of the Poisson equation iterative solver so that the time-marching process can move forward efficiently and effectively. The fluid flow model integrates peripheral circulation using a lumped-parameter model (LPM) to capture the field pressures and flows across various circulatory compartments. Machine learning opens the doors to an intelligent approach for iterative solutions by replacing prescribed criteria with an algorithm that uses the data set itself to predict convergence

Multiphasic model development and meshless simulations of electric-sensitive hydrogels

Master'sMASTER OF ENGINEERIN

Consistency and convergence of SPH approximations

Includes bibliographical references (leaves 58-59).Includes abstract.This thesis is about a new approach to SPH. Instead of using a single kernel or shape function for approximation of a function and its derivatives, individual shape functions are used for each derivative. The investigation is carried out in one space dimension. After producing the conditions for consistency and convergence for the zeroth, first and second derivatives, a new set of linear or piecewise-linear shape functions which meet the minimum of these requirements are presented for each

Development and applications of the Finite Point Method to compressible aerodynamics problems

This work deals with the development and application of the Finite Point Method (FPM) to compressible aerodynamics problems. The research focuses mainly on investigating the capabilities of the meshless technique to address practical problems, one of the most outstanding issues in meshless methods. The FPM spatial approximation is studied firstly, with emphasis on aspects of the methodology that can be improved to increase its robustness and accuracy. Suitable ranges for setting the relevant approximation parameters and the performance likely to be attained in practice are determined. An automatic procedure to adjust the approximation parameters is also proposed to simplify the application of the method, reducing problem- and user-dependence without affecting the flexibility of the meshless technique. The discretization of the flow equations is carried out following wellestablished approaches, but drawing on the meshless character of the methodology. In order to meet the requirements of practical applications, the procedures are designed and implemented placing emphasis on robustness and efficiency (a simplification of the basic FPM technique is proposed to this end). The flow solver is based on an upwind spatial discretization of the convective fluxes (using the approximate Riemann solver of Roe) and an explicit time integration scheme. Two additional artificial diffusion schemes are also proposed to suit those cases of study in which computational cost is a major concern. The performance of the flow solver is evaluated in order to determine the potential of the meshless approach. The accuracy, computational cost and parallel scalability of the method are studied in comparison with a conventional FEM-based technique. Finally, practical applications and extensions of the flow solution scheme are presented. The examples provided are intended not only to show the capabilities of the FPM, but also to exploit meshless advantages. Automatic hadaptive procedures, moving domain and fluid-structure interaction problems, as well as a preliminary approach to solve high-Reynolds viscous flows, are a sample of the topics explored. All in all, the results obtained are satisfactorily accurate and competitive in terms of computational cost (if compared with a similar mesh-based implementation). This indicates that meshless advantages can be exploited with efficiency and constitutes a good starting point towards more challenging applications.En este trabajo se aborda el desarrollo del Método de Puntos Finitos (MPF) y su aplicación a problemas de aerodinámica de flujos compresibles. El objetivo principal es investigar el potencial de la técnica sin malla para la solución de problemas prácticos, lo cual constituye una de las limitaciones más importantes de los métodos sin malla. En primer lugar se estudia la aproximación espacial en el MPF, haciendo hincapié en aquéllos aspectos que pueden ser mejorados para incrementar la robustez y exactitud de la metodología. Se determinan rangos adecuados para el ajuste de los parámetros de la aproximación y su comportamiento en situaciones prácticas. Se propone además un procedimiento de ajuste automático de estos parámetros a fin de simplificar la aplicación del método y reducir la dependencia de factores como el tipo de problema y la intervención del usuario, sin afectar la flexibilidad de la técnica sin malla. A continuación se aborda el esquema de solución de las ecuaciones del flujo. La discretización de las mismas se lleva a cabo siguiendo métodos estándar, pero aprovechando las características de la técnica sin malla. Con el objetivo de abordar problemas prácticos, se pone énfasis en la robustez y eficiencia de la implementación numérica (se propone además una simplificación del procedimiento de solución). El comportamiento del esquema se estudia en detalle para evaluar su potencial y se analiza su exactitud, coste computacional y escalabilidad, todo ello en comparación con un método convencional basado en Elementos Finitos. Finalmente se presentan distintas aplicaciones y extensiones de la metodología desarrollada. Los ejemplos numéricos pretenden demostrar las capacidades del método y también aprovechar las ventajas de la metodología sin malla en áreas en que la misma puede ser de especial interés. Los problemas tratados incluyen, entre otras características, el refinamiento automático de la discretización, la presencia de fronteras móviles e interacción fluido-estructura, como así también una aplicación preliminar a flujos compresibles de alto número de Reynolds. Los resultados obtenidos muestran una exactitud satisfactoria. Además, en comparación con una técnica similar basada en Elementos Finitos, demuestran ser competitivos en términos del coste computacional. Esto indica que las ventajas de la metodología sin malla pueden ser explotadas con eficiencia, lo cual constituye un buen punto de partida para el desarrollo de ulteriores aplicaciones.Postprint (published version

Development and applications of the finite point method to compressible aerodynamics problems

This work deals with the development and application of the Finite Point Method (FPM) to compressible aerodynamics problems. The research focuses mainly on investigating the capabilities of the meshless technique to address practical problems, one of the most outstanding issues in meshless methods. The FPM spatial approximation is studied firstly, with emphasis on aspects of the methodology that can be improved to increase its robustness and accuracy. Suitable ranges for setting the relevant approximation parameters and the performance likely to be attained in practice are determined. An automatic procedure to adjust the approximation parameters is also proposed to simplify the application of the method, reducing problem- and user-dependence without affecting the flexibility of the meshless technique. The discretization of the flow equations is carried out following wellestablished approaches, but drawing on the meshless character of the methodology. In order to meet the requirements of practical applications, the procedures are designed and implemented placing emphasis on robustness and efficiency (a simplification of the basic FPM technique is proposed to this end). The flow solver is based on an upwind spatial discretization of the convective fluxes (using the approximate Riemann solver of Roe) and an explicit time integration scheme. Two additional artificial diffusion schemes are also proposed to suit those cases of study in which computational cost is a major concern. The performance of the flow solver is evaluated in order to determine the potential of the meshless approach. The accuracy, computational cost and parallel scalability of the method are studied in comparison with a conventional FEM-based technique. Finally, practical applications and extensions of the flow solution scheme are presented. The examples provided are intended not only to show the capabilities of the FPM, but also to exploit meshless advantages. Automatic hadaptive procedures, moving domain and fluid-structure interaction problems, as well as a preliminary approach to solve high-Reynolds viscous flows, are a sample of the topics explored. All in all, the results obtained are satisfactorily accurate and competitive in terms of computational cost (if compared with a similar mesh-based implementation). This indicates that meshless advantages can be exploited with efficiency and constitutes a good starting point towards more challenging applications

Meshless Investigation for Nonlocal Elasticity: Static and Dynamic

PhDThe numerical treatment of nonlocal problems, which taking into account material microstructures, by means of meshless approaches is promising due to its efficiency in addressing integropartial differential equations. This thesis focuses on the investigation of meshless methods to nonlocal elasticity. Firstly, mathematical constructions of meshless shape functions are introduced and their properties are discussed. Shape functions based upon different radial basis function (RBF) approximations are implemented and solutions are compared. Interpolation errors of different meshless shape functions are examined. Secondly, the Point Collocation Method (PCM), which is a strong-form meshless method, and the Local Integral Equation Method (LIEM) that bases on the weak-form, are presented. RBF approximations are employed both in PCM and LIEM. The influences of support domains, different kinds of RBFs and free parameters are studied in PCM. While in LIEM, analytical forms of integrals, which is new in meshless method, is addressed. And, the number of straight lines that enclose the local integral domain as well as the integral radius are analyzed. Several examples are conducted to demonstrate the accuracy of PCM and LIEM. Besides, comparisons are made with Abaqus solutions. Then, PCM and LIEM are applied to nonlocal elastostatics based on the Eringen’s model. Formulations of both methods are reported in the nonlocal frame. Numerical examples are presented and comparisons between solutions obtained from both methods are made, validating the accuracy and effectiveness of meshless methods for solving static nonlocal problems. Simultaneously, the influence of characteristic length and portion factors are investigated. Finally, LIEM is employed to solve nonlocal elastodynamic problems. The Laplace transform method and the time-domain technique are implemented in LIEM respectively as the time marching schemes. Numerical solutions of both approaches are compared, showing reasonable agreements. The influence of characteristic length and portion factors are investigated in nonlocal dynamic cases as well.China Scholarship Counci

Research on real-time physics-based deformation for haptic-enabled medical simulation

This study developed a multiple effective visuo-haptic surgical engine to handle a variety of surgical manipulations in real-time. Soft tissue models are based on biomechanical experiment and continuum mechanics for greater accuracy. Such models will increase the realism of future training systems and the VR/AR/MR implementations for the operating room