Category: Theories
-
Open Source Modelling and Visualization Libraries
Introduction Open source 3D modeling and visualization refers to the use of software and tools that are freely available and can be modified and distributed by anyone. These tools and libraries provide a wide range of functionality for 3D modeling and visualization, including the ability to create and edit 3D models, perform numerical simulations, and…
-
NGSOLVE Tutorial (4) Static Condensation Part B. Inhomogeneous Dirichlet B.C and Automatic utility
Inhomogeneous Dirichlet Boundary Conditions In case of inhomogeneous Dirichlet boundary conditions, we combine the technique of Dirichlet data extension with the above static condensation principle in the following code. The code you provided is solving a Poisson equation with a manufactured solution and then calculates the error between the manufactured solution and the numerical solution…
-
NGSOLVE Tutorial (4) Static Condensation Part A
Descriptions Static condensation is a technique used in the solution of partial differential equations (PDEs) to reduce the size of the system to be solved. In the context of the ngsolve example, it likely refers to the use of static condensation to reduce the size of the system of equations being solved by the solver…
-
Polynomial Eigenvalue Problems
A polynomial(high-order) eigenvalue problem is a variation of the standard eigenvalue problem, where the equation $Ax=\lambda x$ is replaced by a polynomial equation of the form where $A_0,A_1,…,A_n$ are matrices and n is an integer greater than 2. The problem is to find the scalars $\lambda$ and non-zero vectors $x$ that are solutions of this…
-
Generalized Eigenvalue Problem
How to Solve Generalized Eigenvalue Problem A generalized eigenvalue problem is a variation of the standard eigenvalue problem, where the matrix $A$ is replaced by two matrices $A$ and $B$. The problem is to find the scalars $\lambda$ and non-zero vectors $x$ such that the equation holds. The matrix $B$ is called the generalized eigenvalue…
-
Eigenvalue Problems
In linear algebra, the eigenvalue problem is a fundamental problem that involves finding the eigenvalues and eigenvectors of a matrix. Given a square matrix $A$, an eigenvalue $\lambda$ is a scalar such that for some non-zero vector $v$. The vector $v$ is called an eigenvector of $A$ corresponding to $\lambda$. The eigenvalue problem can be…
-
Proper Generalized Decomposition(PGD)
Proper Generalized Decomposition (PGD) is a mathematical technique that can be used to approximate the solution of partial differential equations (PDEs) by expanding it in a basis of simple functions. It’s a generalization of the Proper Orthogonal Decomposition (POD) method, which can be used when the solution of the PDE is not linear. PGD is…
-
Proper Orthogonal Decomposition(POD)
Proper Orthogonal Decomposition (POD) is a mathematical technique that can be used to extract the most important modes of a system from a set of snapshots of the system’s state. It is a data-driven method and it can be used to identify the most important features of a system regardless of whether the system is…
-
About Modal Analysis with Nonlinear Materials (2) Pros and Cons of Each Methods
Modal Analysis with Pre-stress Modal analysis with pre-stress is a technique that can be used to analyze the natural frequencies and mode shapes of a structure with non-linear material properties. Here are some of the pros and cons of using this method: Pros: Cons: Non-linear static analysis with a reduced-integration element: Harmonic balance method: Direct…
-
About Modal Analysis with Nonlinear Materials (1) Introductions
Modal Analysis using pre-stress from static nonlinear analysis It is possible to solve for the modal behavior of a structure with non-linear material properties using pre-stress options in some finite element software. However, the specific capabilities of the software and the ability to use pre-stress options for non-linear material modal analysis will depend on the…