Maria Augusta Neto

Stabilization Methods for the Integration of DAE in the Presence of Redundant Constraints

The use of multibody formulations based on Cartesian or naturalcoordinates lead to sets of differential-algebraic equations that haveto be solved. The difficulty in providing compatible initial positionsand velocities for a general spatial multibody model and the finiteprecision of such data result in initial errors that must be correctedduring the forward dynamic solution of the system equations of motion.As the position and velocity constraint equations are not explicitlyinvolved in the solution procedure, any integration error leads to theviolation of these equations in the long run. Another problem that isvery often impossible to avoid is the presence of redundant constraints.Even with no initial redundancy it is possible for some systems toachieve singular configurations in which kinematic constraints becometemporarily redundant. In this work several procedures to stabilize thesolution of the equations of motion and to handle redundant constraintsare revisited. The Baumgarte stabilization, augmented Lagrangian andcoordinate partitioning methods are discussed in terms of theirefficiency and computational costs. The LU factorization with fullpivoting of the Jacobian matrix directs the choice of the set ofindependent coordinates, required by the coordinate partitioning method.Even when no particular stabilization method is used, a Newton–Raphsoniterative procedure is still required in the initial time step tocorrect the initial positions and velocities, thus requiring theselection of the independent coordinates. However, this initialselection does not guarantee that during the motion of the system otherconstraints do not become redundant. Two procedures based on the singlevalue decomposition and Gram–Schmidt orthogonalization are revisited forthe purpose. The advantages and drawbacks of the different procedures,used separately or in conjunction with each other and theircomputational costs are finally discussed.

Introduction to Finite Element Method

As discussed in Chap. 1, mechanic problems are governed by a set of partial differential equations that are valid in a certain domain and they needed to be solved for evaluating the stress condition of mechanical components. Although analytic methods can be employed to solve linear problems involving partial differential equations, its use to analyze complex structures may be a difficult or, even, an impossible task. Thus, in this chapter, Hamilton’s principle, which one of the most powerful energy principle, is introduced for the FEM formulation of problems of mechanics of solids and structures. The approach adopted in this chapter is to directly work out the dynamic system equations, after which the static dynamic equations can be easily obtained by simply dropping out the dynamic terms

Generalized Timoshenko modelling of composite beam structures: sensitivity analysis and optimal design

This article describes a new approach to design the cross-section layer orientations of composite laminated beam structures. The beams are modelled with realistic cross-sectional geometry and material properties instead of a simplified model. The VABS (the variational asymptotic beam section analysis) methodology is used to compute the cross-sectional model for a generalized Timoshenko model, which was embedded in the finite element solver FEAP. Optimal design is performed with respect to the layers’ orientation. The design sensitivity analysis is analytically formulated and implemented. The direct differentiation method is used to evaluate the response sensitivities with respect to the design variables. Thus, the design sensitivities of the Timoshenko stiffness computed by VABS methodology are imbedded into the modified VABS program and linked to the beam finite element solver. The modified method of feasible directions and sequential quadratic programming algorithms are used to seek the optimal continuous solution of a set of numerical examples. The buckling load associated with the twist–bend instability of cantilever composite beams, which may have several cross-section geometries, is improved in the optimization procedure.

Effect of different acid solutions on glass/epoxy composites:

Composite materials are exposed to a range of hostile environments with consequent degradation in terms of mechanical properties. However, the bibliography is not sufficient to establish a full knowledge about the effect of hostile environments on mechanical properties. This work intends to study the response to flexural and to low velocity impact of a glass fibre/epoxy composite after immersion in two different acid solutions, hydrochloric acid and sulphuric acid. These solutions affect significantly the flexural and impact strength, however, the hydrochloric acid was responsible for the worst results. The maximum load and elastic energy decreases with the impacts. On the other hand, the maximum displacement and damaged area showed an inverse tendency.

Engineering Computation of Structures: The Finite Element Method

This book presents theories and the main useful techniques of the Finite Element Method (FEM), with an introduction to FEM and many case studies of its use in engineering practice. It supports engineers and students to solve primarily linear problems in mechanical engineering, with a main focus on static and dynamic structural problems. Readers of this text are encouraged to discover the proper relationship between theory and practice, within the finite element method:Practice without theory is blind, but theory without practice is sterile. Beginning with elasticity basic concepts and the classical theories of stressed materials, the work goes on to apply the relationship between forces, displacements, stresses and strains on the process of modeling, simulating and designing engineered technical systems. Chapters discuss the finite element equations for static, eigenvalue analysis, as well as transient analyses. Students and practitioners using commercial FEM software will find this book very helpful. It uses straightforward examples to demonstrate a complete and detailed finite element procedure, emphasizing the differences between exact and numerical procedures.

Buckling Analysis of Laminated Composite Plates Submitted to Compression After Impact

The aim of this work is the experimental evaluation of the residual strength of composite laminates under compression after low velocity impact. Different kinds of delaminations, partial, embedded and through-the-width, were studied. Carbon-fibre reinforced epoxy composite was used and the experimental tests were performed on [0]16, [0, 90, 0, 90]2S and [0i, 90i]s with i=2, 3, 4, 5, using a drop-weight-testing machine. The impacted plates were inspected by the ultrasonic C-Scan to evaluate the size and the shape of the delamination. Results show that the presence of delamination decreases the compressive strength. The stacking sequence and laminate thickness were verified to have a remarkable influence on the value of the residual compressive strength, and play an important role on the observed buckling failure modes.

Finite Element Method for Beams

A beam is a structural member whose geometry is very similar to the geometry of a bar. It is also geometrically a bar of an arbitrary cross-section, by bar it is meant that one of the dimensions is considerably larger than the other two, whose primary function is to support transverse loading. The main difference between the beam and the truss is the type of load that they support. In fact, beams are the most common type of structural component, especially in civil and mechanical engineering. A beam resists to transverse loads mainly through a bending action and, the bending is responsible for compressive longitudinal stresses in one side of the beam and tensile stress on the other beam side. These two regions are separated by the neutral axis in which the stress is zero. The combination of tensile and compressive stresses produces an internal bending moment. Finite element equations for beam-like structures are developed in this chapter.

Effect of geometry on deformation of anterior implant-supported zirconia frameworks: An in vitro study using digital image correlation

PURPOSE To evaluate the effect of geometry on the displacement and the strain distribution of anterior implant-supported zirconia frameworks under static load using the 3D digital image correlation method. METHODS Two groups (n=5) of 4-unit zirconia frameworks were produced by CAD/CAM for the implant-abutment assembly. Group 1 comprised five straight configuration frameworks and group 2 consisted of five curved configuration frameworks. Specimens were cemented and submitted to static load up to 200N. Displacements were captured with two high-speed photographic cameras and analyzed with video correlation system in three spacial axes U, V, W. Statistical analysis was made using the nonparametric Mann-Whitney test. RESULTS Up to 150N loads, the vertical displacements (V axis) were statistically higher for curved frameworks (-267.83±23.76μm), when compared to the straight frameworks (-120.73±36.17μm) (p=0.008), as well as anterior displacements in the W transformed axis (589.55±64.51μm vs 224.29±50.38μm for the curved and straight frameworks), respectively (p=0.008). The mean von Mises strains over the surface frameworks were statistically higher for the curved frameworks under any load. CONCLUSION Within the limitations of this in vitro study, it is possible to conclude that the geometric configuration influences the deformation of 4-unit anterior frameworks under static load. The higher strain distribution and micro-movements of the curved frameworks reflect less rigidity and increased risk of fractures associated to FPDs.

Effect of round curvature of anterior implant-supported zirconia frameworks: finite element analysis and in vitro study using digital image correlation

Abstract Two groups of 4-unit zirconia frameworks were produced by CAD/CAM to simulate the restoration of an anterior edentulous gap supported by 2 implant-abutment assemblies. Group 1 comprised straight configuration frameworks and group 2 consisted of arched frameworks. Specimens were made with the same connector cross-section area and were cemented and submitted to static loads. Displacements were captured with two high-speed photographic cameras and analysed with video correlation system. Frameworks and the implant-abutment assembly were scanned and converted to 3DCAD objects by reverse engineering process. A specimen of each group was veneered and the corresponding 3D geometry was similarly obtained after scanning. Numerical models were created from the CAD objects and the FE analysis was performed on the zirconia frameworks and on the FPDs bi-layered with porcelain (veneered frameworks). Displacements were higher for the curved frameworks group, under any load. The predicted displacements correlated well with the experimental values of the two framework groups, but on the straight framework the experimental vertical displacements were superior to those predicted by the FEA. The results showed that the round curvature of zirconia anterior implant-supported FPDs plays a significant role on the deformation/stress of FPDs that cannot be neglected neither in testing nor in simulation and should be considered in the clinical setting.

Advanced FEM Modelling

This chapter presents a discussion on some modelling techniques for the stress analyses of solids and structures. Mesh symmetry, rigid elements and constraint equations, mesh compatibility, modelling of offsets, supports and connections between elements with different mathematical bases are all covered. Advanced modelling of laminated composite materials are also presented.