An obstacle problem for Koiter’s shells

2019 ◽  
Vol 24 (10) ◽  
pp. 3061-3079 ◽  
Author(s):  
Philippe G Ciarlet ◽  
Paolo Piersanti
Keyword(s):  
Asymptotic Analysis ◽  
A Priori ◽  
Elastic Shell ◽  
Three Dimensional ◽  
Two Dimensional ◽  
Three Dimensional Model ◽  
Limit Model ◽  
Membrane Shells ◽  
Point Of Departure ◽  
Confinement Condition

In this paper, we define, a priori, a natural two-dimensional Koiter’s model of a ‘general’ linearly elastic shell subject to a confinement condition. As expected, this model takes the form of variational inequalities posed over a non-empty closed convex subset of the function space used for the ‘unconstrained’ Koiter’s model. We then perform a rigorous asymptotic analysis as the thickness of the shell, considered a ‘small’ parameter, approaches zero, when the shell belongs to one of the three main classes of linearly elastic shells, namely elliptic membrane shells, generalized membrane shells and flexural shells. To illustrate the soundness of this model, we consider elliptic membrane shells to fix ideas. We then show that, in this case, the ‘limit’ model obtained in this fashion coincides with the two-dimensional ‘limit’ model obtained by means of another rigorous asymptotic analysis, but this time with the three-dimensional model of a ‘general’ linearly elastic shell subject to a confinement condition as a point of departure. In this fashion, our proposed Koiter’s model of a linearly elastic shell subject to a confinement condition is fully justified in this case, even though it is not itself a ‘limit’ model.

An obstacle problem for elliptic membrane shells

2018 ◽  
Vol 24 (5) ◽  
pp. 1503-1529 ◽  
Author(s):  
Philippe G. Ciarlet ◽  
Cristinel Mardare ◽  
Paolo Piersanti
Keyword(s):  
Vector Field ◽  
Asymptotic Analysis ◽  
Obstacle Problem ◽  
Displacement Vector ◽  
Middle Surface ◽  
Two Dimensional ◽  
Displacement Vector Field ◽  
Membrane Shells ◽  
Signorini Condition ◽  
Confinement Condition

Our objective is to identify two-dimensional equations that model an obstacle problem for a linearly elastic elliptic membrane shell subjected to a confinement condition expressing that all the points of the admissible deformed configurations remain in a given half-space. To this end, we embed the shell into a family of linearly elastic elliptic membrane shells, all sharing the same middle surface [Formula: see text], where [Formula: see text] is a domain in [Formula: see text] and [Formula: see text] is a smooth enough immersion, all subjected to this confinement condition, and whose thickness [Formula: see text] is considered as a “small” parameter approaching zero. We then identify, and justify by means of a rigorous asymptotic analysis as [Formula: see text] approaches zero, the corresponding “limit” two-dimensional variational problem. This problem takes the form of a set of variational inequalities posed over a convex subset of the space [Formula: see text]. The confinement condition considered here considerably departs from the Signorini condition usually considered in the existing literature, where only the “lower face” of the shell is required to remain above the “horizontal” plane. Such a confinement condition renders the asymptotic analysis substantially more difficult, however, as the constraint now bears on a vector field, the displacement vector field of the reference configuration, instead of on only a single component of this field.

Clinical experience of using virtual 3D modelling for pre and intraoperative guidance during robotic-assisted partial nephrectomy

2021 ◽  
pp. 205141582110002
Author(s):  
Lorenz Berger ◽  
Aziz Gulamhusein ◽  
Eoin Hyde ◽  
Matt Gibb ◽  
Teele Kuusk ◽  
...  
Keyword(s):  
Computed Tomography ◽  
Partial Nephrectomy ◽  
Surgical Outcome ◽  
Prospective Cohort ◽  
Three Dimensional ◽  
Two Dimensional ◽  
Dimensional Model ◽  
Three Dimensional Model ◽  
Robotic Assisted ◽  
Robotic Assisted Partial Nephrectomy

Objective: Surgical planning for robotic-assisted partial nephrectomy is widely performed using two-dimensional computed tomography images. It is unclear to what extent two-dimensional images fully simulate surgical anatomy and case complexity. To overcome these limitations, software has been developed to reconstruct three-dimensional models from computed tomography data. We present the results of a feasibility study, to explore the role and practicality of virtual three-dimensional modelling (by Innersight Labs) in the context of surgical utility for preoperative and intraoperative use, as well as improving patient involvement. Methods: A prospective study was conducted on patients undergoing robotic-assisted partial nephrectomy at our high volume kidney cancer centre. Approval from a research ethics committee was obtained. Patient demographics and tumour characteristics were collected. Surgical outcome measures were recorded. The value of the three-dimensional model to the surgeon and patient was assessed using a survey. The prospective cohort was compared against a retrospective cohort and cases were individually matched using RENAL (radius, exophytic/endophytic, nearness to collecting system or sinus, anterior/posterior, location relative to polar lines) scores. Results: This study included 22 patients. Three-dimensional modelling was found to be safe for this prospective cohort and resulted in good surgical outcome measures. The mean (standard deviation) console time was 158.6 (35) min and warm ischaemia time was 17.3 (6.3) min. The median (interquartile range) estimated blood loss was 125 (50–237.5) ml. Two procedures were converted to radical nephrectomy due to the risk of positive margins during resection. The median (interquartile range) length of stay was 2 (2–3) days. No postoperative complications were noted and all patients had negative surgical margins. Patients reported improved understanding of their procedure using the three-dimensional model. Conclusion: This study shows the potential benefit of three-dimensional modelling technology with positive uptake from surgeons and patients. Benefits are improved perception of vascular anatomy and resection approach, and procedure understanding by patients. A randomised controlled trial is needed to evaluate the technology further. Level of evidence: 2b

scholarly journals Numerical simulation of the dimensional transformation of atomization in a supersonic aerodynamic atomization dust-removing nozzle based on transonic speed compressible flow

2020 ◽  
Vol 7 (3) ◽  
pp. 597-610 ◽  
Author(s):  
Tian Zhang ◽  
Deji Jing ◽  
Shaocheng Ge ◽  
Jiren Wang ◽  
Xiangxi Meng ◽  
...  
Keyword(s):  
Flow Field ◽  
Axisymmetric Flow ◽  
Three Dimensional ◽  
Two Dimensional ◽  
The Finite Element Method ◽  
Three Dimensional Model ◽  
Speed Flow ◽  
Complex Process ◽  
Study Results ◽  
Laval Nozzles

Abstract To simulate the transonic atomization jet process in Laval nozzles, to test the law of droplet atomization and distribution, to find a method of supersonic atomization for dust-removing nozzles, and to improve nozzle efficiency, the finite element method has been used in this study based on the COMSOL computational fluid dynamics module. The study results showed that the process cannot be realized alone under the two-dimensional axisymmetric, three-dimensional and three-dimensional symmetric models, but it can be calculated with the transformation dimension method, which uses the parameter equations generated from the two-dimensional axisymmetric flow field data of the three-dimensional model. The visualization of this complex process, which is difficult to measure and analyze experimentally, was realized in this study. The physical process, macro phenomena and particle distribution of supersonic atomization are analyzed in combination with this simulation. The rationality of the simulation was verified by experiments. A new method for the study of the atomization process and the exploration of its mechanism in a compressible transonic speed flow field based on the Laval nozzle has been provided, and a numerical platform for the study of supersonic atomization dust removal has been established.

Blast load simulation using conical shock tube systems

2019 ◽  
Vol 11 (2) ◽  
pp. 135-158 ◽  
Author(s):  
Ahmed Ismail ◽  
Mohamed Ezzeldin ◽  
Wael El-Dakhakhni ◽  
Michael Tait
Keyword(s):  
Shock Tube ◽  
Three Dimensional ◽  
Design Parameters ◽  
Civil Infrastructure ◽  
Two Dimensional ◽  
Blast Loads ◽  
Dimensional Model ◽  
Tube System ◽  
Three Dimensional Model ◽  
Conical Shock

With the increased frequency of accidental and deliberate explosions, evaluating the response of civil infrastructure systems to blast loading has been attracting the interests of the research and regulatory communities. However, with the high cost and complex safety and logistical issues associated with field explosives testing, North American blast-resistant construction standards (e.g. ASCE 59-11 and CSA S850-12) recommend the use of shock tubes to simulate blast loads and evaluate relevant structural response. This study first aims at developing a simplified two-dimensional axisymmetric shock tube model, implemented in ANSYS Fluent, a computational fluid dynamics software, and then validating the model using the classical Sod’s shock tube problem solution, as well as available shock tube experimental test results. Subsequently, the developed model is compared to a more complex three-dimensional model and the results show that there is negligible difference between the two models for axisymmetric shock tube performance simulation; however, the three-dimensional model is necessary to simulate non-axisymmetric shock tubes. Following the model validation, extensive analyses are performed to evaluate the influences of shock tube design parameters (e.g. the driver section pressure and length and the expansion section length) on blast wave characteristics to facilitate a shock tube design that would generate shock waves similar to those experienced by civil infrastructure components under blast loads. The results show that the peak reflected pressure increases as the driver pressure increases, while a decrease in the expansion length increases the peak reflected pressure. In addition, the positive phase duration increases as both the driver length and expansion length are increased. Finally, the developed two-dimensional axisymmetric model is used to optimize the dimensions of a physical large-scale conical shock tube system constructed for civil infrastructure component blast response evaluation applications. The capabilities of such shock tube system are further investigated by correlating its design parameters to a range of explosion threats identified by different hemispherical TNT charge weight and distance scenarios.

scholarly journals A Justification of Two-Dimensional Nonlinear Viscoelastic Shells Model

2012 ◽  
Vol 2012 ◽  
pp. 1-24
Author(s):  
Fushan Li
Keyword(s):  
Asymptotic Expansion ◽  
Asymptotic Analysis ◽  
Laplace Transformation ◽  
Three Dimensional ◽  
Two Dimensional ◽  
Nonlinear Viscoelastic ◽  
Leading Term ◽  
Viscoelastic Shells

By applying formal asymptotic analysis and Laplace transformation, we obtain two-dimensional nonlinear viscoelastic shells model satisfied by the leading term of asymptotic expansion of the solution to the three-dimensional equations.

Eclectic modelling of the North Atlantic. II. Transient tracers and the ventilation of the Eastern Basin thermocline

1988 ◽  
Vol 325 (1583) ◽  
pp. 201-236 ◽  
Keyword(s):  
North Atlantic ◽  
Three Dimensional ◽  
Two Dimensional ◽  
Ekman Pumping ◽  
Dimensional Model ◽  
Three Dimensional Model ◽  
Oxygen Utilization ◽  
The North ◽  
Time Histories ◽  
Weak Constraints

Renewal rates of the waters of the thermocline in the eastern North Atlantic are estimated by combining linear quasi-geostrophic dynamics with steady and transient tracers into a unified eclectic, reservoir model. The two-dimensional model first employed is finally rejected when it is found that it generates oxygen-utilization rates (OUR) that are, by conventional biological wisdom, too high. The three-dimensional model that replaces the two-dimensional one shows that the our is indeterminate, with possible ranges from zero to unacceptably high values. The region is flushed primarily from the north and east. The problem of using transient tracers is mathematically equivalent to that of distributed-system boundary-control theory, the open-ocean boundary conditions playing the role of the unknown control variables. The missing time histories of this new set of unknowns means that tritium and helium-3 distributions are only comparatively weak constraints on the flow field, but do set upper bounds on the vertical exchange with surface waters. Surface Ekman pumping is adequate to explain the interior distributions without additional buoyancy ventilation, although this latter process is possible. Some speculation is made about conditions under which transient tracers might play a more definitive role.

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