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# 数学代写|有限元代写Finite Element Method代考|ENEM28001 Assembly of Global FE Equation

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## 数学代写|有限元代写Finite Element Method代考|Assembly of Global FE Equation

The FE equations for all the individual elements can be assembled together to form the global FE system equation:
$$\mathbf{K D}+M \ddot{\mathbf{D}}=\mathbf{F}$$
where $\mathbf{K}$ and $\mathbf{M}$ are the globe stiffness and mass matrix, $\mathbf{D}$ is a vector of all the displacements at all the nodes in the entire problem domain, and $\mathbf{F}$ is a vector of all the equivalent nodal force vectors. The process of assembly is one of simply adding up the contributions from all the elements connected at a node. The detailed process will be demonstrated in Chapter 4 using an example. It may be noted here that the assembly of the global matrices can be skipped by combining assembling with the equation solving. This means that the assembling of a term in the global matrix is done only when the equation solver is operating on this term.

## 数学代写|有限元代写Finite Element Method代考|Imposition of Displacement Constraints

The global stiffness matrix $\mathbf{K}$ in Eq. (3.96) does not usually have a full rank, because displacement constraints (supports) are not yet imposed, and it is non-negative definite or positive semi-definite. Physically, an unconstrained solid or structure is capable of performing rigid movement. Therefore, if the solid or structure is free of support, Eq. (3.96) gives the behavior that includes the rigid body dynamics, if it is subjected to dynamic forces. If the external forces applied are static, the displacements cannot be uniquely determined from Eq. (3.96) for any given force vector. It is meaningless to try to determine the static displacements of an unconstrained solid or structure that can move freely.

For constrained solids and structures, the constraints can be imposed by simply removing the rows and columns corresponding to the constrained nodal displacements. We shall demonstrate this method in an example problem in later chapters. After the treatment of constraints (and if the constraints are sufficient), the stiffness matrix $\mathbf{K}$ in Eq. (3.96) will be of full rank, and will be Positive Definite (PD). Since we have already proven that $\mathbf{K}$ is symmetric, $\mathbf{K}$ is of a Symmetric Positive Definite (SPD) property.

## 数学代写|有限元代写Finite Element Method代考|Assembly of Global FE Equation

$$\mathbf{K D}+M \ddot{\mathbf{D}}=\mathbf{F}$$

## MATLAB代写

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