1. Linear linear, which refers to the relationship between quantity and quantity in proportion and in a straight line. It can be understood mathematically as a function in which the first derivative is a constant;
Non-linear non-linear refers to the relationship that is not proportional and not linear, and the first derivative is not constant.
2, linear can be considered as a curve, such as y = ax + b, that is a straight line
A nonlinear one can be considered as a curve of more than 2 times, such as y=ax^2+bx+c, (x^2 is the power of x), that is, it is not a straight line.
3. The relationship between two variables is a one-time relationship - the image is a straight line, and the relationship between the two variables is a "linear relationship";
If it is not a functional relationship - the image is not a straight line, it is a "nonlinear relationship."
4. "Linear" and "nonlinear" are often used to distinguish the dependence of the function y = f (x) on the independent variable x. A linear function is a linear function whose image is a straight line. Other functions are nonlinear functions whose images are not straight lines.
Linearity, the proportional and linear relationship between quantity and quantity, represents regular and smooth motion in space and time, while nonlinearity refers to non-proportional, non-linear relationship, representing irregular motion and mutation.
For example, a common resistor is a linear component, and the voltage U across the resistor R has a linear relationship with the current I flowing, that is, R=U/I, and R is a constant number. The forward characteristic of a diode is a typical nonlinear relationship. The voltage u across the diode is not a fixed ratio to the current i flowing through it, ie the forward resistance of the diode is a function of the different operating points (u, i). ) and different.
5. In mathematics, the linear relationship means that the independent variable x and the dependent variable yo can be expressed as y=ax+b, (a, b is a constant), that is, a linear relationship between x and y.
Can not be expressed as y = ax + b, (a, b is a constant), that is, a nonlinear relationship, the nonlinear relationship can be a quadratic, cubic or other functional relationship, or it may be irrelevant.
Misunderstanding
1. The difference between linear and nonlinear is whether the sample can be separated by a straight line (this view is correct)
2. Discuss with the classmates whether the logistics model is linear or non-linear, and it is difficult to understand! (the logistics model is a generalized linear model)
3, distinguish between regression and classification problems, linear models can be used for curve fitting (regression), but the classification of linear model models must be a straight line, such as the logistics model.
Difference between linear model and nonlinear model
1. A linear model can fit a sample with a curve, but the decision boundary of the classification must be straight, such as a logistics model.
2. Distinguish whether it is a linear model. The main reason is to look at the coefficient w before the independent variable x in a multiplication formula. If w affects only one x, then the model is a linear model. Or determine if the decision boundary is linear
3. Examples
Draw y and x are curve relationships, but it is a linear model, because x1*w1 can be observed that x1 is only affected by one w1
This model is a nonlinear model. It is observed that x1 is not only affected by the parameter w1 but also by w5. If the independent variable x is affected by more than two parameters, then the model is nonlinear!
4, in fact, the most simple to determine whether a model is linear, only need to determine whether the decision boundary is a straight line, that is, whether it can be divided by a straight line
Neural network is nonlinear
Although each node of the neural network is a logistics model, it is a nonlinear model combined.
Here we only consider the three-layer neural network
First layer expression
Second layer expression
Bringing the expression of the first layer into the second layer expression, we can observe that the x1 variable is not only affected by w1 but also by k2, so this model is not a linear model, it is a nonlinear model.
The difference between linear analysis and nonlinear analysisLinear analysis refers to the elastic part of the stress-strain curve at the beginning, that is, the structural analysis that does not reach the stress yield point. Nonlinear analysis includes state nonlinearity, geometric nonlinearity, and material nonlinearity. State nonlinearity such as fishing rods, geometry such as large deformation of objects, materials such as plastic material properties.
2. Reasons for nonlinear behavior
There are many reasons for causing structural nonlinearity, which can be mainly divided into the following three types.
(1) State changes (including contact)
Many common structures exhibit a state-dependent nonlinear behavior. For example, a cable that can only be stretched may be slack or taut; the bearing sleeve may or may not be in contact; the frozen soil may be frozen or melted. The stiffness of these systems changes abruptly due to changes in system conditions. The state change may be directly related to the load (as in the case of a cable) or it may be caused by some external cause (such as turbulent thermodynamic conditions in frozen soil). Contact is a very common nonlinear behavior, and contact is a special and important subset of the nonlinear types of state changes.
(2) Geometric nonlinearity
If the structure is subjected to large deformations, its varying geometry may cause a nonlinear response of the structure. The fishing rod shown in Figure 5.2 will have a large deformation under a slight load. As the vertical load increases, the rod is continuously bent, resulting in a significant reduction in the power arm, resulting in an increasing stiffness of the rod at higher loads.
(3) Material nonlinearity
Nonlinear stress-strain relationships are a common cause of structural nonlinearities. Many factors can affect the stress-strain properties of a material, including loading history (eg, under elastic-plastic response conditions), environmental conditions (such as temperature), total time of loading (eg, under creep response conditions).
3. Problems that should be paid attention to in nonlinear structural analysis
(1) Newton-Raphson method
The equation solver of the ANSYS program can predict the response of an engineering system by calculating a series of simultaneous linear equations. However, the behavior of nonlinear structures cannot be directly represented by such a series of linear equations, and a series of linear approximations with corrections are needed to solve nonlinear problems.
An approximate nonlinear solution is to divide the load into a series of load increments. The load increment can be applied in several load steps or in several sub-steps of one load step. After each incremental solution is completed, the program adjusts the stiffness matrix to reflect the nonlinear variation of structural stiffness before proceeding to the next load increment. Unfortunately, the pure incremental approximation inevitably accumulates errors with each load increment, and the resulting results are out of balance, as shown in Figure 5.3a.
The ANSYS program overcomes this difficulty by using Newton-Raphson equilibrium iterations, which achieves a balanced convergence of the end solutions for each load increment within a certain tolerance. Figure 5.3b depicts the use of Newton-Raphson equilibrium iterations in a single-degree-of-freedom nonlinear analysis. Before each solution, the NR method estimates the residual vector, which is the difference between the restoring force (the load corresponding to the unit stress) and the applied load. After that, the program uses a non-equilibrium load for linear solution and checks for convergence. If the convergence criterion is not met, the unbalanced load is re-estimated, the stiffness matrix is ​​modified, a new solution is obtained, and the iterative process continues until the problem converges.
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