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Methods To Calculate Wave Propagation

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Question:

Discuss about the Different Methods To Calculate Wave Propagation Analysis.
 
 

Answer:

Introduction

According to (Braja, 2016)there is a need for Civil Engineers to properly understand different theories and analysis that are used to evaluate soils and foundation designs. The extensive Geotechnical Engineering is a wide study that helps mechanics in their day to day study of soil and rocks, learning their features and soil composition. In this report, we are looking at a study of wave propagation analysis of two-phase saturated porous media using coupled finite–infinite element method. An extensive study of soil and its behaviors, the decay function of sub-surface materials is shown based on the analytical solution. A properly managed study of the phenomenon of wave propagation in water bearing media must have effective and results. There are many methods used to deal with unbounded domains. However both methods are good and give estimated results. The finite and infinite domain. Computationally, Differential Equations governs wave propagation, saturated slightly in porous media.

Computational Framework/ Numerical Method.

Computational framework is the basis of observing chemical processes reactions. In the research method of wave propagation analysis of two phase saturated porous media using finite-infinite element method, computational framework/numerical model is the finite method proposed for analyzing the remote domains. This method is usually used in engineering and mathematical physics to come up with solutions of numeric. The finite method is used to truncate boundaries that are at a large distance remotely from one zone then fixed or free boundary conditions are imposed. This approach have its own limitations whereby, if the waves reflect back near the field, wrong results may be incurred. This approach has its own limitations in case the systems used are taken legal action. The finite method may lead to high costs resolving from computational, large storage needed and time frame penalties. These are the major limitations of the numerical method.

Main features of Computational Framework.

One of the features of the truncating approach is imposing a special boundary condition whereby the infinite domain is truncated at an arbitrary location, for example absorption of energy. This method however is not satisfactory because they are mostly artificial.

Another feature of the Framework is using a finite element that are coupled together and boundary element method. This is whereby there is a division of the whole system to the closest field, that in cooperates symmetrical boundaries and non- homogeneous and those fields that extent to limitlessness.

The third feature in the finite method, is using the cloning method that was proposed by known Dasguta. Which was later made better by wolf and Song. The advantage of this method is that it is the only finite approach that is standalone. However its main weakness is whereby, some conditions of similarity of geometry and property of material can be satisfied.

Another great feature for computational framework is the use of finite element and the infinite approach. This is used to measure the infinity in different ways whereby the finite elements measure near area and in-finite one used to measure areas that are far.

 

Other methods used

Governing Equations

Governing equations is whereby, mass is conserved and energy too is conserved in fluid. In wave propagation analysis, the porous media can be compressed with viscid fluids.  

Infinite Element Formulation

This is where shape decays with distance and zero is reaches infinity. The shape functionality does not matter a lot here. This methods consists of two main steps whereby, there is need of analytical identification solution of the problem and derivation of the shape from it. This method in cooperates several solutions.

I.D analytical approach is one of the solutions of the infinite element foundation whereby element functions shapes are derived from it. Shape functions is another type of infinite element solution whereby the shape functions are the key elements. Property functions is another approach whereby all directions are shown.

Finite Element Formulation.

This is where, the research is done using Galerkin approach. The Finite Element Method (FEM) is a numerical technique that is used to get estimated solutions of partial differential equations. FEM, was originated from the need of solving complex elasticity and structural analysis problems in Civil Engineering. It aids in giving strength and stiffness to structures that are being simulated. Moreover assists in cost elimination and weight minimization to structures that are being built. This method subdivides large tasks into smaller parts that are simple to tackle which are called finite elements. (Joonsang, 2012)

Infinite Element Formulation

Infinite element Formulation (IFEM), is usually calculated using integer m, known as the infinite element order. For one to get the smallest error possible between estimated and exact solution, then the order of integer m, should be highest. These elements, the infinite elements are used in acoustic models to represent the radiation of field on finite elements that are unbounded. They have many advantages over some of the boundary treatment of such tasks.  While carrying out this example, providing stability to such structures some of the factors to be taken into consideration entails avoidance of very big, dominant massing, large elongated or slab-like plates, being very innovative and creative with appropriate choice of materials especially key in the, inaccurate methods of computation of stresses and strains from the effects of shrinkage, this is to mean only shapeless materials are used.rimming the infinite domain at an arbitrary location then imposing great boundary locations. This is where shape decays with distance to zero as reaches it infinity. The shape functionality does not matter a lot here. This methods consists of two main steps whereby, there is need of analytical identification solution of the problem and derivation of the shape from it. This method in cooperates several solutions. (Joonsang, 2012)

Verification method

Verification method commonly have four parts, which are intense inspection, demonstration of the results, testing and analyzing the findings. In inspection, the common methods used are usually the five senses which are tasting, touching, seeing, smelling or olfactory. This is used to identify the accuracy and efficiency of the infinite method during wave propagation analysis. Two experiment are carried out and then they are compared to verify the similarities. Example, a problem with 1D problem that consists saturated porous media subjected to a uniform harmonic loading with circular frequency. A schematic representation of the problem and the finite–infinite element the near field is discretized using eight-node isoperimetric finite elements and the far field is modelled using a single infinite element.

Comparison of Finite Element method and other Methods

Theoretically finite element method has more advantages compared to other methods on porous media. The most known advantage is stability. Finite element method is more stable compared to the other methods and easy to establish. It is good to know prior so that one will not use it unknowingly.
 
Convergence is another advantage of finite method because variation forms usually are consistent with governing equations. The approximation of finite method usually follows from best approximate results.
 
The finite method is easily adaptable thus making adaptivity the third advantage of finite method over the others. This is where you have to rely on indication and not estimation.  The other method show where error might be and not the exact place.
 
Computationally, finite method also has some advantages as listed below,     
 
Hybridization this is where the mixed formulation method is used, where you use second order term as systems of two first order terms.    
Inhomogeneity this is when one used higher order quadrature rule in finite method naturally.     
Complex geometrics, this is where infinite method is used to solve problems theoretically given that one has a good mesh generator, without changing a code.   
Boundary conditions, this is whereby finite element method is used to resort conditions that are considered weak.With the above comparisons, of finite elements over the other methods, the advantages make the method seem to be the most efficient method to use in the testing’s.

Conclusion

In conclusion, wave propagation problems have been fully analyzed and it is seen that that have saturated the soils in great way. This includes domains that are unbounded. Geotechnical Engineering should be incorporated more and many approaches used to come up with an accurate answer or method. Application of the infinite element is discussed into length to show efficiency of the proposed element. The Finite method may seem to be the best, but keeping in mind the other methods too are all well perceived. The main aim is to come up with a better method that will give accurate results irrespective of the shape of the soils or surfaces. To conclude, when the two methods are in cooperated or used together, they tend to bring out accurate results, until when infinite elements are introduced and then the numerical results seems to disappear. For Civil Engineers to come up with the best method, they need to test and proof test the method and finally use the one that does not strain, or limit them in any way possible.

 

References

Athanasios, P., & Thomas, B. (2010). Soil Engineering. Berlin: Heidelberg.

Braja, M. (2016). Principles of Foundation Engineering. Australia: Cengage Learning.

Celebi, E., Goktepe, F., & Karahan, N. (2012). Non-linear finite element analysis for ptrediction of seismic response of buildings considering soil-structure interaction. Copemicus GmbH.

Delwyn, G., & Murray, D. (2012). Unsaturated soils mechanics in engineering. Hoboken, N.J: Wiley.

Hao, L. (2008). Diffraction of SH-waves by surface or sub-surface topographies with application to soil-structure interaction on shallow foundations. Los Angeles: California.

Jien, H., & Andrew, J. (2011). Geo-Frontiers 2011: advances in geotechnical Engineering. Reston: VA

Joonsang, P. (2012). Wave motion in finite and infinite media using the thin layer method.

Karl, T., & Ralph, B. (2013). Soil Mechanics in Engineering practice. England: Read Books Ltd.

Lutz, L. (2007). Wave propagation in infinite Domains: with applications to structure interaction. Dordrecht: Springer.

Reddy, R. (2010). Soil Engineering. New Delhi: GeneTech Books.

Rodney, L. (2013). Soil and Water conservation engineering. St. Joseph: Mich.

Sunjay, K. (2017). Fundamentals of Fibre-Reinforced Soil Engineering. Singapore: Springer Singapore.

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