FEM Equations are Commonly Developed Using Direct, Variational-Virtual Work or Weighted Residual Methods

I think that:-

Direct Method Based on physical reasoning and limited to simple cases, this method is worth studying because it enhances physical understanding of the process, It is Based on the B.C for each case

Weighted Residual Method Starting with the governing differential equation, special mathematical operations develop the “weak form” that can be incorporated into a FEM equation. This method is particularly suited for problems that have no variational statement, it is an integral approach in which we integrate the weighted residual of the governing equation.

weak methods such as direct and variation obtained from weighted Integral form. It has two desirable Characteristic.

- It needs to weaker and less continuity dependent variable and often leads to a set of algebraic equations in terms of coefficients.

- Weak form includes the natural boundary conditions and therefore approximate solution is only necessary to satisfy the basic requirements.

Hi

weighted residual shown to be a powerful method in deriving the finite element equation in the past 8 decades. It is used to transform the governing differential equation to a matrix equation. clearly most of the available softwares are based on this method. The main restriction is that the domain should be continuous. For the problems with discontinuity the method fails. That is a reason for introduction of extended finite element method and etc. which used great time and expense. Recently a state based philosophy is proposed which solved the problem of discontinuity and extends the application of the weighted residual to discontinuous domain, and modified the classical finite element to comply with the discontinuous domain.

Residual Method (WRM). An approximation solution ũ(x) is considered which we replace in the DE. The result is: ũ(x)xx+q=R≠0 (6) where R (the residual) is the errors caused by the approximation. Exact solution of the DE will make its residual zero at all points of the definition domain [0,L]. The philosophy of this WRM is based on obtaining an approximation solution so that the average residual across flow area [0, L] assumes the value zero. This will be achieved by the introduction of so-called weight functions (Wi(x)) so that the weighted integral become null: ∫ RWi dx=∫ [ũxx+q]Widx=0 (7) Wi(x) satisfy the following BC Wi(x=0) = 0 ; Wi(x=L) = 0. The basic principle of weighted residual methods is to minimize the residual in weighted integral. The weak form of the WRM is obtained by partial integration in weighted integral (7): ∫ ũx (Wi)xdx - ∫ qWidx=0 (8) This is called the weak form of the WRM due to this lower differentiability requirements compared to the original weighted residual form (7). Weak form allows us to work with continuous approximate functions and natural BC only. By choosing Wi(x)= ϕi(x) from Ritz’s variational method we obtain the GALERKIN-WRM-approach and the linear system of equations is similar to that of RITZ's variational method. The direct method simple as acompare to residual ...

Direct method is easy method and require time less than

weighting residual and weighting residual is complex method

Weak form obtained from weighted Integral form. It has two desirable Characteristic.

- It needs to weaker and less continuity dependent variable and often leads to a set of algebraic equations in terms of coefficients.

- Weak form includes the natural boundary conditions and therefore approximate solution is only necessary to satisfy the basic requirements.

I hope attached book will be helpful.

This two methods help to solve finit element ...using stiffness matrix and simply the solution ..residual more Accurace to simply differential eq.to matrix .

Hi D. Nizar ,Method of Weighted Residuals (MWR) and direct method are two approaches in Finite Element Methods.Direct Method used for the simple cases, this method is worth studying because it enhances physical understanding of the process, It is Based on the B.C for each case .Method of Weighted Residuals (MWR): is an integral approach in which we integrate the weighted residual of the governing equation and obtain the weak formulation. For example if the p.d.e. in operator form is written as: Lu(x,t) = s where L is the differential operator The residual would be R = Lu(x,t) - s Then the weak formulation would be: Integral_sign (w(x,t) R) = Integral_sign (w(x,t) ((Lu(x,t) – s)) …...................……(1) Where w(x,t) is a weight function and 'w(x,t)R' is essentially the ‘weighted residual’.

Direct method is easy method and require time less than

weighting residual and weighting residual is complex method

Weak form obtained from weighted Integral form. It has two desirable Characteristic.

- It needs to weaker and less continuity dependent variable and often leads to a set of algebraic equations in terms of coefficients.

- Weak form includes the natural boundary conditions and therefore approximate solution is only necessary to satisfy the basic requirements.

The direct approach is related to the “direct stiffness method” of structural analysis and it is the easiest to understand when meeting FEM for the first time. The main advantage of this approach is that you can get a feel of basic techniques and the essential concept involved in the FEM formulation without using much of mathematics. However, by direct approach we can solve only simple problems. While Weighted residual method (WRM) is a class of method used to obtain the approximate solution to the differential equations of the form L(φ)+f=0 in D In WRM, we directly work on differential equation of the problem without relying on any variational principle. It is equally suited for linear and nonlinear differential equations.

In applied mathematics, methods of mean weighted residuals (MWR) are methods for solving differential equations. The solutions of these differential equations are assumed to be well approximated by a finite sum of test functions . In such cases, the selected method of weighted residuals is used to find the coefficient value of each corresponding test function. The resulting coefficients are made to minimize the error between the linear combination of test functions, and actual solution, in a chosen norm.

Direct Approach

The direct approach is related to the “direct stiffness method” of structural analysis and it is the easiest to understand when meeting FEM for the first time. The main advantage of this approach is that you can get a feel of basic techniques and the essential concept involved in

the FEM formulation without using much of mathematics. However, by direct approach we can solve only simple problems.

The direct approach is related to the “direct stiffness method” of structural analysis and it is the easiest to understand when meeting FEM for the first time. The main advantage of this approach is that you can get a feel of basic techniques and the essential concept involved in the FEM formulation without using much of mathematics. However, by direct approach we can solve only simple problems. While Weighted residual method (WRM) is a class of method used to obtain the approximate solution to the differential equations of the form L(φ)+f=0 in D In WRM, we directly work on differential equation of the problem without relying on any variational principle. It is equally suited for linear and nonlinear differential equations.

Direct Approach

The direct approach is related to the “direct stiffness method” of structural analysis and it is

the easiest to understand when meeting FEM for the first time. The main advantage of this

approach is that you can get a feel of basic techniques and the essential concept involved in

the FEM formulation without using much of mathematics. However, by direct approach we can

solve only simple problems.

Weighted Residual Method

Weighted residual method (WRM) is a class of method used to obtain the approximate solution

to the differential equations

In WRM, we directly work on differential equation of the problem without relying on any vari-

ational principle. It is equally suited for linear and nonlinear differential equations. Weightedresidual method involves two major steps. In the first step, we assume an approximate solution

based on the general behavior of the dependent variable. The approximate solution is so selected

that it satisfies the boundary conditions . The assumed solution is then substituted in the

differential equation. Since the assumed solution is only approximate, it does not satisfy the

differential equation resulting in an error or what we call a residual. The residual is then made to

vanish in some average sense over the entire solution domain. This procedure results in a system

of algebraic equations. The second step is to solve the system of equations resulting from the

first step subject to the prescribed boundary condition to yield the approximate solution sought.

Hi

during the past decade we developed a differential equation for analysis of damaged beam-like structure which needs the boundary condition of the intact beam for its solution. This led to great saviving in analysis of these structures. Then the “method of weighted residual” is used to derive the corresponding finite equation. The efforts failed. It is found that the standard method of weighted residual is not prepared for this problem. perhaps this failure is a reason for introduction of the “method of extended finite element“. We extended our effort which finally led to a simple modification in the method of weighted residual and then the finite element method without going into the lengthy and extended finite element method. The by product of these effort was giving birth to the state based philosophy which solved several engineering problem exactly with 1000 times easier than the common rules. More information on the philosophy can be found in researchgate, our publications, and future notes here.

The main difference between weighted residual methods and finite element methods is in the choice of trial function or the shape functions. Traditionally the weighted residual methods have used trail functions which are defined over the entire domain , whereas finite element methods have used shape functions defined over an element, with elements joined together to cover the entire domain . The use of the residual allows for the automatic placement of elements,so that the elements are smallest in regions dictated by the solution . Such a scheme is not feasible in the finite element method when linear shape functions are used , since the residual is not defined .

Direct Approach, is the simplest method for solving discrete problems in 1 and 2 dimensions and it is a powerful method to perfom linear static analysis where the deformation is very small Direct stiffness method is based on applying the force/torque equilibrium to a set of 1D elements enterconnected at nodes to obtain a set of linear algebraic equation, these eq. can be solved for simple problems while the Weighted Residuals method is more complicated and uses the governing differential equations directly (e.g. the Galerkin method), It is equally suited for linear and nonlinear differential equations.

Direct stiffness method:

• The direct stiffness method or (matrix stiffness method) is the most common implementation of the finite element method (FEM). And the system must be modeled as a set of simpler, idealized elements interconnected at the nodes.

• Direct stiffness method is limited for simple 1D problems.But the FEM can be applied to many engineering problems that are governed by a differential equation, so that need systematic approaches to generate FE equations, by using (Weighted residual method).

• Ordinary differential equation (second-order or fourth-order) can be solved using the weighted residual method, in particular using Galerkin method.

Direct method is limited for simple 1D problems while FEM can be applied to many engineering problems that are governed by a differential equation.

Method of Weighted Residuals (MWR): is an integral approach in which we integrate the weighted residual of the governing equation and obtain the weak formulation. The method of weighted residuals (MWR) is an approximate technique for solving boundary value problems that utilizes trial functions satisfying the prescribed boundary conditions and an integral formulation to minimize error, in an average sense, over the problem domain.