Formal Event-B Modeling of the MICONIC Application

2021 ◽  
Author(s):  
Sabrine Ammar ◽  
Mohamed Tahar Bhiri
Keyword(s):  
Dynamic Analysis ◽  
Error Detection ◽  
Posteriori Error ◽  
Formal Proof ◽  
Verification And Validation ◽  
Formal Approach ◽  
Mathematical Proofs ◽  
Automatic Planning ◽  
Refinement Strategy ◽  
Planning Problems

Automatic planning has a de facto standard language called PDDL for describing planning problems. The dynamic analysis tools associated with this language do not allow sufficient verification and validation of PDDL descriptions. Indeed, these tools, namely planners and validators, allow a posteriori error detection. In this paper, we recommend a formal approach coupling the two languages Event-B and PDDL. Event-B supports a formal development process based on the refinement technique with mathematical proofs. Thus, we propose a refinement strategy for obtaining reliable PDDL descriptions from an ultimate Event-B model that is correct by construction. The correctness is guaranteed via the verification and validation tools supported by Event-B. We have chosen the MICONIC application managing modern elevators to illustrate our approach while recognizing that the MICONIC application is already modeled in PDDL without formal proof of its correctness.

Adaptive simulation of free surface flows with discrete least squares meshless (DLSM) method using a posteriori error estimator

2012 ◽  
Vol 29 (8) ◽  
pp. 794-813 ◽  
Author(s):  
G. Shobeyri ◽  
M.H. Afshar
Keyword(s):  
Free Surface ◽  
Posteriori Error ◽  
Free Surface Flows ◽  
Error Estimator ◽  
A Posteriori ◽  
Posteriori Error Estimator ◽  
A Posteriori Error ◽  
Content Type ◽  
Adaptive Simulation ◽  
Refinement Strategy

PurposeThe purpose of this paper is to propose an adaptive refinement strategy based on a posteriori error estimate for the efficient simulation of free surface flows using discrete least squares meshless (DLSM) method.Design/methodology/approachA pressure projection method is employed to discretize the governing equations of mass and momentum conservation in a Lagrangian form. The semi‐discretized equations are then discretized in space using the DLSM method, in which the sum of squared residual of the governing equations and their boundary conditions are minimized with respect to the unknown nodal parameters.FindingsSince the position of the free surface is of great significant in free surface problems, a posteriori error estimator which automatically associates higher error to the nodes near the free surface is proposed and used along with a node moving refinement strategy to simulate the free surface problems more efficiently. To test the ability and efficiency of the proposed adaptive simulation method, two test problems, namely dam break and evolution of a water bubble, are solved and the results are presented and compared to those of analytical and experimental results.Originality/valueError estimate and adaptive refinement have been mostly used in confined and steady‐state flow. Here in this paper, a new attempt has been made to use these concepts in moving boundary problem.

MATHEMATICAL RIGOR AND PROOF

2019 ◽  
pp. 1-41 ◽  
Author(s):  
YACIN HAMAMI
Keyword(s):  
Mathematical Knowledge ◽  
Mathematical Proof ◽  
Philosophy Of Mathematics ◽  
Mathematical Practice ◽  
Formal Proof ◽  
Major Obstacle ◽  
Standard View ◽  
Precise Formulation ◽  
Mathematical Proofs ◽  
Primary Form

Abstract Mathematical proof is the primary form of justification for mathematical knowledge, but in order to count as a proper justification for a piece of mathematical knowledge, a mathematical proof must be rigorous. What does it mean then for a mathematical proof to be rigorous? According to what I shall call the standard view, a mathematical proof is rigorous if and only if it can be routinely translated into a formal proof. The standard view is almost an orthodoxy among contemporary mathematicians, and is endorsed by many logicians and philosophers, but it has also been heavily criticized in the philosophy of mathematics literature. Progress on the debate between the proponents and opponents of the standard view is, however, currently blocked by a major obstacle, namely, the absence of a precise formulation of it. To remedy this deficiency, I undertake in this paper to provide a precise formulation and a thorough evaluation of the standard view of mathematical rigor. The upshot of this study is that the standard view is more robust to criticisms than it transpires from the various arguments advanced against it, but that it also requires a certain conception of how mathematical proofs are judged to be rigorous in mathematical practice, a conception that can be challenged on empirical grounds by exhibiting rigor judgments of mathematical proofs in mathematical practice conflicting with it.

APOS analysis on cognitive process in mathematical proving activities

2018 ◽  
Vol 1 (1) ◽  
pp. 1
Author(s):  
Syamsuri Syamsuri ◽  
Indiana Marethi
Keyword(s):  
Cognitive Process ◽  
Formal Proof ◽  
College Level ◽  
Mathematical Concepts ◽  
Apos Theory ◽  
Mathematical Proofs ◽  
Learning Mathematics ◽  
Partial Type ◽  
Thinking Process ◽  
Mathematical Proving

Thinking is very necessary in learning mathematics, both at school and college level. Several studies have attempted to reveal students' thinking in learning mathematics at college. This article aims to describe the mental structure that occurs when constructing mathematical proofs in terms of APOS theory. The APOS theory has been widely used in analyzing the formation of mathematical concepts in universities. This research explores a thinking process in proof constructing. It uses a qualitative approach. The research was conducted on 26 students majored in mathematics education in public university at Banten, Indonesia. The consideration of that was because the students were able to think a formal proof in mathematics. Results show that there are two types of thinking process in mathematical proving activities, namely:  the deductive-holistic and the inductive-partial type of thinking process. Based on the results, some suitable learning activities should be designed to support the construction of these mental categories.

A POSTERIORI ERROR ESTIMATION AND H-ADAPTIVE REFINEMENT TECHNIQUES FOR TRANSIENT DYNAMIC ANALYSIS OF STIFFENED PLATE/SHELL PANELS

2004 ◽  
Vol 04 (02) ◽  
pp. 259-277
Author(s):  
G. S. PALANI ◽  
NAGESH R. IYER ◽  
T. V. S. R. APPA RAO
Keyword(s):  
Dynamic Analysis ◽  
Posteriori Error ◽  
Adaptive Refinement ◽  
Error Estimator ◽  
A Posteriori ◽  
Transient Dynamic Analysis ◽  
A Posteriori Error ◽  
Transient Dynamic ◽  
Time Stepping ◽  
Adaptive Time

This paper presents a posteriori error estimation and h-adaptive refinement techniques for transient dynamic analysis of stiffened plates/shells using the finite element method (FEM). We furnish the formulation of stiffness and mass matrices for finite element (FE) models, QL9S2 and QUAD4S2 for dynamic analysis of plates/shells with arbitrarily-located concentric/eccentric stiffeners. Procedures for computing a posteriori errors for spatial and temporal errors have been presented for transient dynamic problems. An h-adaptive refinement strategy for stiffened plate/shell panels by employing QL9S2 and QUAD4S2 FE models has been discussed. An adaptive time stepping scheme, which is to be used with the time errors for quality control of the time steps, has also been presented. Numerical studies have been conducted to evaluate the efficacy of the error estimators and the adaptive mesh refinement and time stepping algorithm. The spatial error estimator for transient dynamic analysis is found to exhibit monotonic convergence at all time steps. The temporal error estimator for transient dynamic analysis in association with the adaptive time stepping is able to compute more accurate and reliable time steps to keep the time errors within the specified tolerance limits.

scholarly journals Adaptive IGAFEM with optimal convergence rates: Hierarchical B-splines

2017 ◽  
Vol 27 (14) ◽  
pp. 2631-2674 ◽  
Author(s):  
Gregor Gantner ◽  
Daniel Haberlik ◽  
Dirk Praetorius
Keyword(s):  
Convergence Rates ◽  
Linear Convergence ◽  
Posteriori Error ◽  
Error Estimator ◽  
B Splines ◽  
Energy Error ◽  
Dimension Formula ◽  
Arbitrary Space ◽  
Refined Meshes ◽  
Refinement Strategy

We consider an adaptive algorithm for finite element methods for the isogeometric analysis (IGAFEM) of elliptic (possibly non-symmetric) second-order partial differential equations in arbitrary space dimension [Formula: see text]. We employ hierarchical B-splines of arbitrary degree and different order of smoothness. We propose a refinement strategy to generate a sequence of locally refined meshes and corresponding discrete solutions. Adaptivity is driven by some weighted residual a posteriori error estimator. We prove linear convergence of the error estimator (respectively, the sum of energy error plus data oscillations) with optimal algebraic rates. Numerical experiments underpin the theoretical findings.

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