32 research outputs found
Quadrature Rules for Functions with a Mid-point Logarithmic Singularity in the Boundary Element Method based on the x=tp Substitution
Quadrature rules for evaluating singular integrals that typically occur in the boundary element method (BEM) for two-dimensional and axisymmetric threedimensional problems are considered. This paper focuses on the numerical integration of the functions on the standard domain [-1,1], with a logarithmic singularity at the centre. The subtitution x=tp, where p(>3) is an odd integer is given particular attention, since this returns a regular integral and the domain unchanged. Gauss-Legendre quadrature rules are applied to the transformed integrals for a number of values of p. It is shown that a high values for p typically gives more accurate results
Chatbots put to the test in math and logic problems: A preliminary comparison and assessment of ChatGPT-3.5, ChatGPT-4, and Google Bard
A comparison between three chatbots which are based on large language models,
namely ChatGPT-3.5, ChatGPT-4 and Google Bard is presented, focusing on their
ability to give correct answers to mathematics and logic problems. In
particular, we check their ability to Understand the problem at hand; Apply
appropriate algorithms or methods for its solution; and Generate a coherent
response and a correct answer. We use 30 questions that are clear, without any
ambiguities, fully described with plain text only, and have a unique, well
defined correct answer. The questions are divided into two sets of 15 each. The
questions of Set A are 15 "Original" problems that cannot be found online,
while Set B contains 15 "Published" problems that one can find online, usually
with their solution. Each question is posed three times to each chatbot. The
answers are recorded and discussed, highlighting their strengths and
weaknesses. It has been found that for straightforward arithmetic, algebraic
expressions, or basic logic puzzles, chatbots may provide accurate solutions,
although not in every attempt. However, for more complex mathematical problems
or advanced logic tasks, their answers, although written in a usually
"convincing" way, may not be reliable. Consistency is also an issue, as many
times a chatbot will provide conflicting answers when given the same question
more than once. A comparative quantitative evaluation of the three chatbots is
made through scoring their final answers based on correctness. It was found
that ChatGPT-4 outperforms ChatGPT-3.5 in both sets of questions. Bard comes
third in the original questions of Set A, behind the other two chatbots, while
it has the best performance (first place) in the published questions of Set B.
This is probably because Bard has direct access to the internet, in contrast to
ChatGPT chatbots which do not have any communication with the outside world
Investigation of the Static Bending Response of FGM Sandwich Plates
In the present work, a displacement-based high-order shear deformation theory is introduced for the static response of functionally graded plates. The present theory is variationally consistent and strongly similar to the classical plate theory in many aspects. It does not require the shear correction factor, and gives rise to the transverse shear stress variation so that the transverse shear stresses vary parabolically across the thickness to satisfy free surface conditions for the shear stress. By dividing the transverse displacement into the bending and shear parts and making further assumptions, the number of unknowns and equations of motion of the present theory is reduced a and hence makes them simple to use. The material properties of the plate are assumed to be graded in the thickness direction according to a simple power-law distribution in terms of volume fractions of material constituents. The equilibrium equations of a functionally graded plate are given based on the higher order shear deformation theory. The numerical results presented in the paper are demonstrated by comparing the results with solutions derived from other higher-order models found in the literature and the present numerical results of Finite Element Analysis (FEA). In the numerical results, the effects of the grading materials, lay-up scheme and aspect ratio on the normal stress, shear stress and static deflections of the functionally graded sandwich plates are presented and discussed. It can be concluded that the proposed theory is accurate, elegant and simple in solving the problem of the bending behavior of functionally graded plates
Dynamic response of rigid wheels on deformable terrains
Off-road vehicle performance, such as vehicle mobility, maneuverability, and traction performance is generally affected by the pneumatic tyre-off-road terrain interaction. Modelling of such cases is usually based on empirical and semi-empirical solutions, which have limited applicability in real situations due to their inherent weaknesses. In this study, numerical simulation of the dynamic mobility of a rigid wheel on a deformable terrain is performed through a series of transient nonlinear dynamic finite element analyses with the use of the finite element code ABAQUS (v. 6.13). The dynamic interaction of a rigid wheel with the underlying soil during off-road vehicle travel is simulated. The effects of the vertical load carried by the wheel, the tread pattern, the longitudinal and lateral tread parameters, and the slip ratio of the wheel on the wheel performance are investigated and useful results are extracted. The numerical results reveal that the effects of the tread pattern, particularly tread depth and the terrain constitutive properties, such as soil cohesion can be of high importance for the general wheel response
Development of accurate pneumatic tyre finite element models based on an optimization procedure
A novel method for extracting the geometric and constitutive material properties of pneumatic tyres from available numerical or experimental data for the development of realistic and reliable tyre numerical models is proposed. This method involves an optimization procedure, which incorporates a finite element model as a solver (ABAQUS) properly coupled with an optimiser function (MATLAB). Following that, an initial tyre model (P235/75R17) is developed, and then its properties are suitably adjusted via the optimization process, in order for the former to best fit a target model available in the literature, with respect to eigenfre-quency analysis results. After the termination of the algorithm, the “optimum” tyre model (i.e. the model which best conforms to the target model) is obtained, the response of which is further investigated to ensure its realistic behaviour, which warrants its use for various numerical simulations. The results of this study show clearly the efficiency of the optimization procedure proposed, as well as the realistic response of the tyre model developed
Finite element modelling of a pneumatic tyre interacting with rigid road and deformable terrain
In this study, the dynamic interaction between a pneumatic P235/75R17 tyre and a rigid and/or deformable rolling surface is analysed using the finite element method. After a footprint analysis of the tyre, two
models were developed: (a) tyre rolling on rigid road with bumps and (b) tyre rolling on the surface of a soft soil layer, appropriate for off-road vehicle handling. In the first model, a steady-state transport analysis is performed to obtain the free rolling conditions of the tyre, which are then used to simulate the impact of the tyre on a rigid bump, typical of a road cleat or speedbump. In the second model, the transient rolling response of the tyre on a soft cohesive soil layer was simulated under towed and driven conditions. Along with a
concise literature review and detailed description of the numerical models
considered, various response quantities along with deformed geometries are
provided throughout the text. It is finally shown that the dynamic response of the rolling tyre and its interaction with the deformable soil is inherently complex and depends on many parameters, whereas it is proven that the overall performance of the off-road tyres depends on significantly different parameters from that of the on-road tyres
Pneumatic tyres interacting with deformable terrains
In this study, a numerical model of a deformable tyre interacting with a deformable
road has been developed with the use of the finite element code ABAQUS (v. 6.13). Two tyre models with different widths, not necessarily identical to any real industry tyres, have been created purely for research use. The behaviour of these tyres under various vertical loads and different inflation pressures is studied, initially in contact with a rigid surface and then with a
deformable terrain. After ensuring that the tyre model gives realistic results in terms of the interaction with a rigid surface, the rolling process of the tyre on a deformable road was studied. The effects of friction coefficient, inflation pressure, rebar orientation and vertical load on the overall performance are reported. Regarding the modelling procedure, a sequence of models were analysed, using the coupling implicit – explicit method. The numerical results
reveal that not only there is significant dependence of the final tyre response on the various initial driving parameters, but also special conditions emerge, where the desired response of the tyre results from specific optimum combination of these parameters
Dynamic response of rigid wheels on deformable terrains
Off-road vehicle performance, such as vehicle mobility, maneuverability, and traction performance is generally affected by the pneumatic tire-off-road terrain interaction. Modeling of such cases is usually based on empirical and semi-empirical solutions, which have limited applicability in real situations due to their inherent weaknesses. In this study, numerical simulation of the dynamic mobility of a rigid wheel on a deformable terrain is performed through a series of transient nonlinear dynamic finite element analyses with the use of the finite element code ABAQUS (v. 6.13). The dynamic interaction of a rigid wheel with the underlying soil during off-road vehicle
travel is simulated. The effects of the vertical load carried by the wheel, the tread pattern, the longitudinal and lateral tread parameters, and the slip ratio of the wheel on the wheel performance are investigated and useful results are extracted. The numerical results reveal that the effects of the tread pattern particularly tread depth and the terrain constitutive properties, such as soil cohesion can be of high importance for the general wheel response
Off-road tire-terrain interaction: an analytical solution
A novel semi-analytical solution has been developed for the calculation of the static and dynamic response of an off road tire interacting with a deformable terrain, which utilizes soil parameters independent of the size of the contact patch (size-independent). The models involved in the solution presented, can be categorized in rigid and/or pneumatic tires, with or without tread pattern. After a concise literature review of related methods, a detailed presentation of the semi-analytical solution is presented, along with assumptions and limitations. A flowchart is provided, showing the main steps of the numerical implementation, and various test cases have been examined, characterized in terms of vertical load, tire dimensions, soil properties, deformability of the tire, and tread pattern. It has been found that the proposed model can qualitatively capture the response of a rolling wheel on deformable terrain