BME402 Computational Methods in Engineering Analysis Assignment Sample NUIG Ireland

BME402 is an introductory-level course in computational methods for engineering analysis. The course covers a range of mathematical methods that can be used to solve engineering problems, including linear algebra, numerical analysis, and scientific computing. In addition, students learn how to use software tools for engineering analysis, including MATLAB and Python.

The course is designed for students in engineering programs who want to learn how to use computers to solve real-world problems. It provides a foundation in mathematical methods and software tools that students can build on as they continue their studies in engineering.

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In this section, we are describing some assigned activities. These are:

Assignment Activity 1: Develop the finite element equations from potential energy or other functional statement governing the process.

To develop the finite element equations from potential energy, we must first start with the statement governing the process. This states that the potential energy of a system is equal to the sum of its kinetic and internal energies. From this, we can derive the following equation:

U = K + E

where U is potential energy, K is kinetic energy, and E is internal energy. By taking the derivative of both sides concerning time, we obtain:

dU/dt = DK/dt + dE/dt

Now, if we assume that the system is in equilibrium (i.e., no net forces are acting on it), then we can further simplify this equation to:

dU/dt = 0

Since the potential energy of a system is equal to the sum of its kinetic and internal energies, we can set each term in the equation equal to zero and solve for U. This gives us:

U = -K – E

where U is potential energy, K is kinetic energy, and E is internal energy.

Now that we have the potential energy equation, we can develop the finite element equations from it. To do this, we first need to discretize the equation. This means that we need to divide the equation into a series of smaller equations, each representing a small element of the system. For example, if we were considering a beam, we would divide it into a series of small sections, each representing a small element of the beam.

Assignment Activity 2: Develop suitable interpolation functions for the formulation of one-dimensional, two-dimensional, and axisymmetric elements.

There are a variety of interpolation functions that can be used for one-dimensional, two-dimensional, and axisymmetric elements. A few popular options include linear interpolation, cubic spline interpolation, and quintic polynomial interpolation.

Each type of interpolation function has its benefits and drawbacks. Linear interpolation is simple and fast, but it can produce poor results if the data points are not evenly spaced. Cubic spline interpolation produces smoother results than linear interpolation but is slightly more computationally expensive. Quintic polynomial interpolation produces the smoothest results of all three options but is also the most computationally expensive.

The best interpolation function to use depends on the specific problem you are trying to solve. In general, linear interpolation is sufficient for most problems, but you may want to use cubic spline or quintic polynomial interpolation if your data points are not evenly spaced or if you need very smooth results.

Assignment Activity 3: Apply finite element solution techniques to problems in solid mechanics.

There are numerous ways to apply finite element solution techniques to problems in solid mechanics, and the specific approach used will depend on the nature of the problem being solved. In general, however, the finite element method can be used to approximate the solution to a complex problem by breaking it down into a series of smaller, simpler problems that can be solved individually. This allows for a more accurate overall solution than other methods, such as the review Rayleigh-Ritz method or the calculus of variations.

One common way to use the finite element method is to discretize a continuous domain into a set of small elements, often using triangular or quadrilateral shapes. The equations governing the behaviour of these elements are then solved simultaneously to obtain the desired solution.

Another way to use the finite element method is to discretize a differential equation into a set of algebraic equations. This can be done by using the Galerkin method, which involves multiplying the differential equation by a test function and integrating it over the domain. This results in a system of linear equations that can be solved to obtain the desired solution.

The finite element method can also be used to solve problems in which the domain is not continuous, such as problems with discontinuous boundaries or discontinuous material properties. In these cases, the domain is discretized into a set of small subdomains, and the equations governing the behaviour of each subdomain are solved independently. The results from each subdomain are then combined to obtain the overall solution.

Assignment Activity 4: Demonstrate a knowledge of the implementation of the finite element method in a computer program.

The finite element method (FEM) is a numerical analysis technique used to solve problems that are difficult or impossible to solve using other methods. In a FEM program, the problem is divided into several small parts or elements, and each element is solved individually. This technique can be used to solve problems in physics, engineering, and mathematics.

The finite element method was first developed in the 1950s by Dr Richard Courant at New York University. Today, it is one of the most popular methods for solving problems in engineering and physics. It has been used to develop computer programs for the simulation of structures such as bridges, aeroplanes, and cars, as well as products such as golf clubs and tennis rackets.

The finite element method is also used in the analysis of problems in solid mechanics, such as the stress and strain in a material. In this application, the problem is divided into several small elements, and each element is solved individually. The results from each element are then combined to obtain the overall solution.

Assignment Activity 5: Demonstrate an ability to model and solve a range of practical problems, using the Abaqus software suite, covering the areas of elasticity, plasticity, contact, and heat conduction.

The Abaqus software suite is a powerful tool for modelling and solving a wide range of practical problems in the areas of elasticity, plasticity, and contact. It can be used to simulate the behaviour of engineering systems under various loading conditions and can help engineers understand the response of real-world materials to forces acting on them.

The software suite can also be used to predict the performance of new or proposed designs, and optimize existing designs. In addition, Abaqus can be employed to investigate failure modes and develop prevention strategies. Ultimately, the use of Abaqus can lead to better-engineered products and safer working conditions.

Assignment Activity 6: Make use of finite element techniques in other project and design exercises.

Finite element techniques can be applied in a variety of ways to improve project and design exercises. For example, they can be used to help analyze structural integrity, optimize designs for weight and performance, or predict strain and stress on components. In addition, FEA can also help identify potential failure points in a design early on in the development process, which can save time and money later on.

Overall, using FEA can be an extremely valuable tool for improving project and design outcomes. By taking into account various factors such as stress and strain levels, engineers and designers can create more durable and reliable products that perform better under strenuous conditions.

Assignment Activity 7: Develop the finite volume equations for mass, energy, and momentum conservation.

The finite volume method is a discretization technique for partial differential equations that result in a system of algebraic equations. This approach is attractive because it leads to stable schemes with relatively little effort. Additionally, the long-term behaviour of the numerical solution can be studied using energy methods, leading to some insight into neural networks’ stability behaviour. We will present the development of the finite volume equations for mass, energy, and momentum balance conservation. 

We start with an open control volume V that contains some continuum like water or air. The intention is to idealize the control volume as small as possible so that we can neglect physical effects that might come from its boundaries or curvature. The OFV should be taken such that it has smooth boundaries. A control volume can be visualized as a 3D box with a small opening at the top, as shown in Figure 1.

The control volume is partitioned into several small sub-control volumes using internal surfaces S. The size of the sub-control volumes will be denoted by ΔV. The outward-pointing normal vector to the surface of each sub-control volume is denoted by n̂. With this notation, the control volume can be written as V=ΣΔV.

Assignment Activity 8: Select suitable boundary conditions, discretization techniques, and solution methods for 2D and 3D steady and transient problems.

There are a variety of methods that can be used to select suitable boundary conditions, discretization techniques, and solution techniques for 2D and 3D steady and transient problems. Some of the more common methods include:

• Finite difference methods
• Finite element methods
• Boundary element methods
• Finite volume methods 

The choice of method depends on the type of problem being solved and the desired level of accuracy. Finite difference methods are typically used for simple problems where a high degree of accuracy is not required, while finite element and boundary element methods are often used for more complex problems where a higher degree of accuracy is desired. In general, the higher the degree of accuracy required, the more complex and time-consuming the solution process will be.

Assignment Activity 9: Apply computational fluid dynamics (CFD) solution techniques to problems in the fluids systems.

CFD is a powerful tool that can be used to simulate fluid flow and identify potential issues in fluid systems. By understanding the principles of CFD, engineers can design more efficient and effective fluid systems. In this article, we will discuss how to apply computational fluid dynamics (CFD) solution techniques to problems in the fluids systems.

Computational fluid dynamics (CFD) is a branch of mechanics that deals with the analysis and prediction of fluid behaviour using numerical methods. CFD is used extensively in many industries such as aerospace, automotive, chemical, environmental, naval architecture, and biomedical engineering. The goal of using CFD is to understand how a particular system interacts with its environment and optimize its performance.

CFD is used to simulate fluid flow and heat transfer in systems. The benefits of using CFD include the ability to study a wide range of problems, the ability to obtain results quickly, and the ability to obtain accurate results. Additionally, CFD can be used to study a wide range of phenomena including laminar and turbulent flow, compressible and incompressible flow, and convective and diffusive heat transfer.

Assignment Activity 10: Demonstrate a knowledge of the implementation of CFD methods in a computer program.

CFD methods can be implemented in a computer program using different numerical techniques. The most common approach is the finite difference method, which discretizes the governing equations into a set of algebraic equations that can be solved numerically. Other approaches include the finite element method and the finite volume method.

The first step in implementing CFD methods in a computer program is to discretize the governing equations. This process involves breaking up the continuous derivatives in the equations into discrete forms. For example, if we are solving the heat equation, we would discretize the Temperature derivative concerning space and time:

T(x,t) = T^n_i+1 – 2T^n_i + T^n_i-1 / Δx2 + γT^n+1_i – 2γT^n_i + γT^n-1_i / Δt

where n is the time step, i is the spatial node, and γ is the Courant number.

After the governing equations have been discretized, the next step is to write a computer program to solve the resulting algebraic equations. There are a variety of ways to do this, but the most common approach is to use an iterative method such as the Gauss-Seidel method or the Jacobi method.

Once the computer program has been written, it can be used to simulate fluid flow and heat transfer in a variety of systems. For example, CFD can be used to study the flow of air around an aeroplane wing or the flow of water through a pipe. Additionally, CFD can be used to optimize the performance of a system by adjusting the design variables.

Assignment Activity 11: Demonstrate an ability to model and solve a range of practical problems, using the ANSYS CFD software suite, covering the areas of single-phase flow, mixing, convection heat transfer, and diffusion.

The ANSYS CFD software suite can be used to model and solve a range of practical problems in the areas of single-phase flow, mixing, and chemical engineering. With over three decades of experience in fluid dynamics simulation, ANSYS has a proven track record of delivering reliable and accurate results.

The ANSYS CFD software suite is based on the Finite Element Method (FEM), which allows for the solution of complex engineering problems using a mesh of discrete elements. This mesh can be tailored to represent the geometry and physics of the problem being solved, making it possible to capture even the most subtle effects. In addition, the ANSYS CFD software suite features an extensive range of solvers and Physical Models that can be used to simulate a wide variety of engineering problems.

Assignment Activity 12: Make use of CFD techniques in other project and design exercises.

CFD techniques can be used in several different ways to improve the design and engineering of a variety of projects. One way that CFD can be used is by analyzing the airflow around an object. This information can then be used to improve the aerodynamics of the object, making it more efficient and reducing drag.

Another way that CFD can be used is in the simulation of heat transfer. This can help optimize exhaust systems or identify potential problems with cooling systems. Additionally, CFD can also be used to simulate fluid dynamics, which is important for projects such as water treatment plants or oil refineries. Overall, CFD provides a wealth of data that can help engineers improve their designs and create more efficient systems.

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