Energy consistent nonlinear dynamic contact analysis of structures

This work is motivated by the need for a numerically stable dynamic contact algorithm, for use with finite element (FE) analysis including both material and geometric nonlinearities, which imposes the appropriate full kinematic compatibility between the interfaces of impacting boundaries during a persistent dynamic contact. Several methods were previously developed based on Lagrangian multipliers or penalty functions in an attempt to impose the impenetrability condition of dynamic contact analysis. Some of these existing algorithms suffer from lack of numerical stability, and most of them are incapable of accurately predicting the persistent contact force, hence they would not be suitable for frictional dynamic contact analysis. The numerical stability and energy conservation characteristics of conventional frictionless dynamic contact algorithms using Lagrangian displacement constraints and penalty functions are investigated in this thesis. Two energy controlling dynamic contact algorithms are proposed in conjunction with the well-known Newmark trapezoidal rule, namely, regularised penalty method and Lagrangian velocity constraint. Although energy consistent, the state of the art for these two methods is somewhat similar to the conventional displacement constraints in the sense that acceleration compatibility is not imposed when simulating problems featuring persistent dynamic contact. In this work, a novel and superior energy controlling-algorithm is proposed which overcomes the aforementioned shortcomings. The proposed DVA method enforces the displacement, velocity and acceleration compatibilities (referred to as DVA constraint in this work) between the impacting interfaces, which in contrast to existing algorithms can be used for FE analysis of problems exhibiting geometric and material nonlinearities. The advanced DVA method is devised such that the kinematic compatibilities at the interface are consistent with the solution for a continuous system without any special treatment in the time-integration or solution procedure of the penetrating interface boundaries. Furthermore, this can be achieved in conjunction with all of the prevalent implicit time-integration schemes such as the trapezoidal rule, midpoint rule, HHT-α and the most recently developed Energy-Momentum family of Methods. Finally, utilising the proposed dynamic contact algorithms, a novel multi-constraints node-to-surface dynamic contact element is formulated and programmed within a geometric and material nonlinear dynamic FE analysis software. Several verification examples of frictionless mechanical contact are presented to demonstrate the superiority and performance of the developed node-to-surface contact element in conjunction with the proposed DVA constraint as well as the Lagrangian velocity constraint, providing a robust and accurate solution procedure for highly nonlinear dynamic contact analysis.Open Acces

Differential-Algebraic Equations and Beyond: From Smooth to Nonsmooth Constrained Dynamical Systems

The present article presents a summarizing view at differential-algebraic equations (DAEs) and analyzes how new application fields and corresponding mathematical models lead to innovations both in theory and in numerical analysis for this problem class. Recent numerical methods for nonsmooth dynamical systems subject to unilateral contact and friction illustrate the topicality of this development.Comment: Preprint of Book Chapte

On the constraints violation in forward dynamics of multibody systems

It is known that the dynamic equations of motion for constrained mechanical multibody systems are frequently formulated using the Newton-Euler’s approach, which is augmented with the acceleration constraint equations. This formulation results in the establishment of a mixed set of partial differential and algebraic equations, which are solved in order to predict the dynamic behavior of general multibody systems. The classical resolution of the equations of motion is highly prone to constraints violation because the position and velocity constraint equations are not fulfilled. In this work, a general and comprehensive methodology to eliminate the constraints violation at the position and velocity levels is offered. The basic idea of the described approach is to add corrective terms to the position and velocity vectors with the intent to satisfy the corresponding kinematic constraint equations. These corrective terms are evaluated as function of the Moore-Penrose generalized inverse of the Jacobian matrix and of the kinematic constraint equations. The described methodology is embedded in the standard method to solve the equations of motion based on the technique of Lagrange multipliers. Finally, the effectiveness of the described methodology is demonstrated through the dynamic modeling and simulation of different planar and spatial multibody systems. The outcomes in terms of constraints violation at the position and velocity levels, conservation of the total energy and computational efficiency are analyzed and compared with those obtained with the standard Lagrange multipliers method, the Baumgarte stabilization method, the augmented Lagrangian formulation, the index-1 augmented Lagrangian and the coordinate partitioning method.The first author expresses his gratitude to the Portuguese Foundation for Science and Technology through the PhD grant (PD/BD/114154/2016). This work has been supported by the Portuguese Foundation for Science and Technology with the reference project UID/EEA/04436/2013, by FEDER funds through the COMPETE 2020 – Programa Operacional Competitividade e Internacionalização (POCI) with the reference project POCI-01-0145-FEDER-006941.info:eu-repo/semantics/publishedVersio

Design and Analysis of Air-Stiffened Vacuum Lighter-Than-Air Structures

Lighter-than-air (LTA) systems have been developed for numerous applications and have taken several forms. Airships, aerostats, blimps, and balloons are all part of this family of systems, which uses Archimedes principle to achieve neutral and positive buoyancy in air by replacing an air volume with LTA gases. These lifting gases stiffen the otherwise compliant envelope structures, allowing them to sustain the pressure difference brought by the displaced air. The compliance of these structures is a byproduct of the weight requirement, materials and geometrical arrangement of which these structures are built from, typically resulting in dimensionalities that exhibit low or virtually non-existent in-plane bending stiffness. The former has constrained the development of LTA structures that utilize an internal partial vacuum, rather than a lifting gas, to achieve positive buoyancy, where the structure would be subjected to a pressure differential near atmospheric pressure. Given the above limitation, this research presents the development trajectory and structural characterization of air stiffened designs, which utilize air to shape and serve as the core of a set of enclosing envelopes. The development trajectory established a simulation framework that enables the structural characterization of air-stiffened designs under a variety of geometric and loading conditions. Such framework allowed for the development of finite element solutions that included geometric, fluid-structure and contact nonlinearities, with capacity for further generalization. Given the developed framework, the structural characterization of the Helical Sphere and Icoron air-stiffened designs demonstrated a reduction of material modulus and strength requirements compared to membrane-over-frame designs, and showed the capability of air-stiffened designs to be tailored for specific material strength limits

A Variational Level Set Approach for Surface Area Minimization of Triply Periodic Surfaces

In this paper, we study triply periodic surfaces with minimal surface area under a constraint in the volume fraction of the regions (phases) that the surface separates. Using a variational level set method formulation, we present a theoretical characterization of and a numerical algorithm for computing these surfaces. We use our theoretical and computational formulation to study the optimality of the Schwartz P, Schwartz D, and Schoen G surfaces when the volume fractions of the two phases are equal and explore the properties of optimal structures when the volume fractions of the two phases not equal. Due to the computational cost of the fully, three-dimensional shape optimization problem, we implement our numerical simulations using a parallel level set method software package.Comment: 28 pages, 16 figures, 3 table

Variational Methods for Biomolecular Modeling

Structure, function and dynamics of many biomolecular systems can be characterized by the energetic variational principle and the corresponding systems of partial differential equations (PDEs). This principle allows us to focus on the identification of essential energetic components, the optimal parametrization of energies, and the efficient computational implementation of energy variation or minimization. Given the fact that complex biomolecular systems are structurally non-uniform and their interactions occur through contact interfaces, their free energies are associated with various interfaces as well, such as solute-solvent interface, molecular binding interface, lipid domain interface, and membrane surfaces. This fact motivates the inclusion of interface geometry, particular its curvatures, to the parametrization of free energies. Applications of such interface geometry based energetic variational principles are illustrated through three concrete topics: the multiscale modeling of biomolecular electrostatics and solvation that includes the curvature energy of the molecular surface, the formation of microdomains on lipid membrane due to the geometric and molecular mechanics at the lipid interface, and the mean curvature driven protein localization on membrane surfaces. By further implicitly representing the interface using a phase field function over the entire domain, one can simulate the dynamics of the interface and the corresponding energy variation by evolving the phase field function, achieving significant reduction of the number of degrees of freedom and computational complexity. Strategies for improving the efficiency of computational implementations and for extending applications to coarse-graining or multiscale molecular simulations are outlined.Comment: 36 page

Une méthode mixte multi-échelles pour un simulateur de réservoir biphasé

A multiscale hybrid mixed finite element method is presented in this paper to solve two-phase flow equations on heterogeneous media under the effect of gravitational segregation. It is designed to cope with the complex geometry and inherent multiscale nature of the rocks, leading to stable and accurate multi-physics reservoir simulations. This multiscale approach makes use of coarse scale fluxes between subregions (macro domains) that allow to reduce substantially the dominant computational costs associated with the flux/pressure kernel embedded in the numerical model. As such, larger scale problems can be approximated in a reasonable computational time. Dividing the problems into macro domains leads to a hierarchy of meshes and approximation spaces, allowing the efficient use of static condensation and parallel computation strategies. The method documented in this work utilizes discretizations based on a general domain partition formed by poly-hedral subregions. The normal flux between these subregions is associated with a finite dimensional trace space. The global system to be solved for the fluxes and pressures is expressed only in terms of the trace variables and of a piecewise constant pressure associated with each subregion. The fine scale features are resolved by mixed finite element approximations using fine flux and pressure representations inside each subregion, and the trace variable (i.e. normal flux) as Neumann boundary conditions. This property implies that the flux approximation is globally H(div)-conforming, and, as in classical mixed formulations, local mass conservation is observed at the micro-scale elements inside the subregions, an essential property for flows in heterogeneous media