Using Finite Element Methods to Calculate the Deflection of an Orifice Plate Subject to Uniform Pressure Distribution

As part of an elective course in Finite Element Methods (FEM) for senior level and graduate students in mechanical engineering, an ASME standard for flow measurement devices is used to design an orifice plate. Students are given a certain set of flow condition and equipment constraints that they must adhere to. As part of the design process, they are required to evaluate their orifice plate for strength via finite element methods and determine if the plate’s transverse deflections due to uniformly distributed pressure are within set limits. To design the orifice plate, a symbolic solver (Wolfram Mathematica) is used to solve the governing fourth order differential equation of this problem (plate equation in polar coordinates), with appropriate boundary conditions. Results from the symbolic solver are juxtaposed with results from a GUI/Menu driven FEM package (Altair Hyperworks suite). Both the symbolic and menu driven solutions are compared with each other and with published relationships. Governing equations for bending of plates, in polar coordinates (for the orifice plate) have the need to resolve mathematical singularities for “1/r”, for “r=0” type terms. This when reconciled using symbolic solvers allows a better grasp of the esoteric inter-relationships between various terms in the governing equations, which are akin to design variables. This allows students to use this esoteric knowledge to better apply GUI/menu driven solvers for engineering design. The primary pedagogical goal of this work allows the exertion of importance of governing equation based modeling to improve a “behind the scenes” understanding of GUI/menu driven FEM efforts. Students are made aware of the use of engineering standards and validation of numerical solutions based on numerical accuracy and convergence of solution through comparison with analytical data. Introduction and Philosophy Modern day FEM is closed attached to the advent of mathematical and matrix algebra methods in the design of aeronautical structures1,2,3,4. Primarily, FEM is a method of approximations to solve field problems. The power of FEM packages is realized when the fundamental field problems governing the engineering design are “encompassed” in irregular shapes. In this paper, regular shapes are: square/rectangular geometrics, circular cross sections. In the combined undergraduate and graduate level mechanical engineering course on “Introduction to Finite Element Methods”, the aim of this ongoing work is to temper the use of commercial FEM packages with a sound understanding of fundamental engineering physics and differential equations (classical analysis). This allows for an emphasis on the strengths and limitations of the software package and the analytical solutions and methods to make good FEM pre-processing decisions. An exposure to analytical methods also allows students to design experiments/technology and to analyse and interpret results and data obtained effectively. To do this, a project is introduced in designing an orifice plate (standard flow measuring device) through the use of a commercial FEM package (the Hyperworks suite) with result validation obtained from analytical solutions from the Theory of Elasticity (the Biharmonic equation is used). Our university is an ABET accredited university. The exercise described in this paper is in accordance with ABET’s 2015-2016 criteria for accrediting engineering programs. ABET’s criteria states “The curriculum must require students to apply principles of engineering, basic science, and mathematics (including multivariate calculus and differential equations); to model, analyze, design, and realize physical systems, components or processes; and prepare students to work professionally in either thermal or mechanical systems while requiring topics in each area.” The following student outcomes are outlined by ABET in their criteria for accrediting engineering programs and they will be referred to as ABET (a)-(k) as is customarily the case: (a) An ability to apply knowledge of mathematics, science, and engineering (b) An ability to design and conduct experiments, as well as to analyze and interpret data (c) An ability to design a system, component, or process to meet desired needs within realistic constraints such as economic, environmental, social, political, ethical, health and safety, manufacturability, and sustainability (d) an ability to function on multidisciplinary teams (e) An ability to identify, formulate, and solve engineering problems (f) An understanding of professional and ethical responsibility (g) An ability to communicate effectively (h) The broad education necessary to understand the impact of engineering solutions in a global, economic, environmental, and societal context (i) A recognition of the need for, and an ability to engage in life-long learning (j) A knowledge of contemporary issues (k) An ability to use the techniques, skills, and modern engineering tools necessary for engineering practice. The objectives (primary and supplemental), successes and failures of this study are described briefly in the project assessment table 1. Objective Success Reason for Failure Comments Use of analytical equations and symbolic solvers to solve the engineering problem Yes n/a n/a Validation of results obtained from FEM package with analytical solution Yes n/a n/a Validation of results obtained from FEM package with experimental results No Limited experimental data found by students. Students need to start acquiring data at earlier stage. Supplementary goal of communication of scientific results Yes n/a ASME style extended abstracts were authored by students Supplementary goal of identifying other engineering problems with same fundamental core Yes n/a During this project, students uncovered many applications (machine components) of the annular plate and some pursued these alternate design problems. Table 1: Project Assessment Table. Since this paper describes a work in progress, some supplementary goals were added. Organization of Paper In the succeeding sections, we discuss the use and relevance of symbolic solvers and how it allows for development of new skills by students. The evolution of the design project that students are to complete and the design project itself are described. Sample results of FEM package (Hyperworks Suite) driven analysis and symbolic solver (Wolfram Mathematica) are described and compared. This discussion of results also includes a brief description of analytical modeling that is an input to the symbolic solver. Finally, the pedagogic results with respect to engineering simulation are discussed along with a brief history of the fundamental differential equation (Biharmonic equation) at the core of the technology being designed in this. Use of Symbolic Solvers Symbolic solvers (also known as “Computer Algebra Systems”) automate the solution of systems of linear equations through the interpretation of symbols (coefficients, independent variables etc.). The application of symbolic solvers has had a dramatic, positive impact on science and technology with various accounts of its effectiveness and application available5,6,7,8 and its usage in education as a primary or complementary portion to enhance student understanding of concepts9,10,11,12. These solvers lend themselves exceptionally well to classical analysis of engineering problems. In our FEM course, Wolfram Mathematica is chosen for classical analysis because of its powerful differential equation solver, NDSolve. The author has tested higher order, non-linear, stiff spatiotemporal differential equations in engineering analysis through the use Mathematica’s NDSolve function and results have been highly favorable. The author has focussed NDSolve on numerical or analytical solutions of non-linear partial differential equations as part of his research experience. Since symbolic solvers allow for the interpretation of both numbers and symbols, their versatility is greater than traditional procedural programming techniques such as FORTRAN, C/C++. They do possess some drawbacks in the learning curve associated with them and that they traditionally use JAVA based interpreters unlike the compiler driven FORTRAN, C/C++. However, for the solution of fundamentally important differential equations in elasticity, fluid mechanics, heat transfer (among many fields), symbolic solvers allow for a greater ease in obtaining numerical or analytical solutions than the more time-consuming procedural programming methods. It is this niche that is explored by mechanical engineering students in this FEM course, through the obtention of closed form analytical solutions using symbolic solvers. Closed form analytical solutions allow for an intimate understanding of the relationship between various terms in a differential equation and their weights and how these terms relate to the physics being captured. The weights of differential equations are generally design (structural or fluid) parameters or properties. Some classical examples are provided in equation 1 and equation 2. ∂u ∂t = α ( ∂u ∂x2 + ∂u ∂y2 + ∂u ∂z2 ) + q̇ ρcp (1) ∂u ∂t2 = c ( ∂u ∂x2 ) (2) The heat equation 1 results from an application of the conservation of energy and Fourier’s law of conduction. The parameters α, ρ, cp are thermal properties of the material through which heat propagation is being studied. These are essentially weights for the second order heat diffusion term or the volumetric heat generation term. The magnitudes of these terms can change the nature of the physics being studied by changing the weight of the second order-in-space effect vs the first order-in-time effect. The wave equation 2 is a result of Newton’s second law. The parameter c is the velocity of propagation of a wave. In case this equation is applied to an axial prismatic bar element, then c is a measure of a stress pulse that travels through the bar element. Again the value of this parameter has an eff