Response model for the psychological education of college students based on nonlinear finite element equations

The data comes from the psychological health survey of college students in the second half of 2.19. The total sample consists of 3558 cases. Among them, 2287 cases (64.3%) are males, and 1271 cases (35.7%) are females. Their ages ranged from 16 to 25 years (average age: 19.5 years).

We use the ‘College Student Personality Inventory’ (University Personality Inventory [UPI]) revised by Professor Fan Fumin of Tsinghua University as the testing tool. The scale consists of 60 items. This includes four false tests and 56 symptoms [2]. All items are 0/1 binary variables. The test items are symptoms that occurred within 1 year. These include the ‘depressive symptoms’ factor and the ‘light business mind’ project. Reliability test of the scale showed that Cronbach’s α = 0.843 and standardised Cronbach’s α = 0.851. This shows that the reliability of the scale is relatively high.

The paper uses the SLSI compiled by Gadzella, which uses the revised scale to detect 51 life stresses of college students in the past 3 months. All items have a five-point value (‘1’ means none, ‘2’ means rarely, ‘3’ means occasionally, ‘4’ means often and ‘5’ means always). The scale is divided into two categories comprising nine factors. The five stressors include frustration, conflict, stress, change and self-imposition. The four factors of the stress response are physiological response, emotional response, behavioural response and cognitive response. The reliability of the original scale shows a value of Cronbach’s α = 0.76. The reliability of the revised scale of the domestic professor Wang Xin shows a value of Cronbach’s α = 0.71. This study tests the reliability of the scale, yielding Cronbach’s α = 0.892 and standardised Cronbach’s α = 0.895. This shows that the reliability of the scale is relatively high.

We perform regression analysis using the Statistical Package for the Social Sciences software version 13 (SPSS13.0) to fit the early-warning model. The article uses Amos4.0 structural analysis to fit the structural equation model and analyses the direct and indirect influence paths. The balanced equation of the non-linear finite element system is as follows: (1) ψ(σ)=P(σ)−R=0 \psi (\sigma) = P(\sigma) - R = 0 (2) P(σ)=∑ceT∫eBTσdV P(\sigma) = \sum c_e^T\int_e {B^T}\sigma dV (3) R=∑ceT∫eNTpdV+∑ceT∫SeNTqdS R = \sum c_e^T\int_e {N^T}pdV + \sum c_e^T\int_{Se} {N^T}qdS

Here, P(σ) and are the equivalent nodal forces of the element stress vector σ and load vectors p,q, respectively; N and c_e are the shape function and selection matrix of the finite element, respectively; B is the conversion matrix between element strain and nodal displacement; ψ(σ) = 0 means that the element stress and external load are in equilibrium; ψ(σ) ≠ 0 means that the element is in an unbalanced state. We call −ψ(σ) = R − P(σ) the unbalanced force vector. The article assumes that the total number of degrees of freedom of the finite element system is N; then, the system node displacement vector a is an N-dimensional vector: (4) a=[a1,a2,⋯,aN]T a = {[{a_1},{a_2}, \cdots,{a_N}]^T}

The equivalent nodal forces P(σ) and R are also N-dimensional vectors. If the stress σ in Eqs (1) and (2) is expressed as strain ε, then, we express it as displacement [3]. In this way, a balanced equation with displacement a as a variable can be obtained: (5) ψ(a)=P(a)−R=0 \psi (a) = P(a) - R = 0

The equivalent nodal force P(σ) and stress σ have a linear relationship. In the case of linear elastic materials, the relationship between P(a) and a is also linear. P(a) is a non-linear function of a. Eq. (5) is given in the complete form. We usually need to use the Newton iterative algorithm for numerical solution. Newton iteration has the characteristic of local convergence. This requires that the initial displacement value a⁰ given by the iteration be closer to the proper solution to ensure that the iterative solution sequence converges to the proper solution. We introduce the load parameter λ and embed it in Eq. (5) to obtain the balance equations with the parameter λ: (6) ψ(a,λ)=P(a)−λR=0 \psi (a,\lambda) = P(a) - \lambda R = 0

Usually, the parameter λ is specified in advance and gradually increases; we discretise it as follows: (7) {0=λ0<λ1<λ2<…<λm<λm+1<…<λM=1Δλ=Δλm=λm+1−λm \left\{{\matrix{{0 = {\lambda _0} < {\lambda _1} < {\lambda _2} < \ldots < {\lambda _m} < {\lambda _{m + 1}} < \ldots < {\lambda _M} = 1} \hfill \cr {\Delta \lambda = \Delta {\lambda _m} = {\lambda _{m + 1}} - {\lambda _m}} \hfill \cr}} \right.

Here, m is the number of load steps applied, and Δλ is the increment of the load parameter. With the increase of λ, we gradually solve the nodal displacement a_m+1 under λ_m+1 for each increment Δλ, which is called the load increment method of solving Eq. (6). In each incremental step, we use the Newton iteration format. Its initial value is taken as the solution a_m of the previous incremental step [4]. As long as the incremental step is small enough, the local convergence condition will be satisfied. The Jacobi matrix (i.e. the tangent stiffness matrix of the system) and the Newton iteration formula of Eq. (6) are, respectively, represented as follows: (8) J=∂ψ∂a=∑ceT∫eBTDepBdVce J = {{\partial \psi} \over {\partial a}} = \sum c_e^T\int_e {B^T}{D_{ep}}BdV{c_e} (9) {Δa0=0Δan+1=Δan−J−1ψ(am+Δan) \left\{{\matrix{{\Delta {a^0} = 0} \hfill \cr {\Delta {a^{n + 1}} = \Delta {a^n} - {J^{- 1}}\psi ({a_m} + \Delta {a^n})} \hfill \cr}} \right.

Here, D_ep is the elastoplastic constitutive matrix. The Jacobi matrix is non-singular (detJ ≠ 0). The iteration can converge to the proper solution a_m+1 in the incremental step Δλ. When m = M, λ_M = 1, the solution of the original system of Eq. (5) is obtained. Introducing the load factor λ and using the incremental method to solve the non-linear equations is the idea of the continuation method. The load increment method is the early (or classic) continuation method.

We used SLSI five-factor life stressor and four-factor stress response as independent variables and UPI depression symptoms as dependent variables for multiple linear stepwise regression analysis. Two early-warning models were established based on the statistically significant (p<0.01) independent variables that entered the equation at the end.

1) The life stressor factors that induce depressive symptoms include frustration, stress, change and self-imposition (with conflict factors being excluded). Beta coefficient shows that frustration has the top influence on depressive symptoms.

We use UPI light business mindfulness (‘0’: None/’1’: Yes) as the dependent variable and depressive symptoms as the covariate to conduct binary logistic regression analysis. The results showed that depressive symptoms significantly affected light business thoughts (p<0.01). The relative risk or rate of light business thoughts for students with depressive symptoms is 1.86 times the odds ratio (OR) of students with non-depressive symptoms. Another group comparison found that the male group’s risk rate of depressive symptoms was 1.84 times the risk rate of non-depressive symptoms. In the female group, the risk rate of depressive symptoms producing light business thoughts was 1.98 times that of non-depressive symptoms [6]. The results are shown in Table 2. At the same time, the predictive classification table generated by the logistic analysis showed that there were 3414 cases with no depressive symptoms and no light business thoughts and 20 cases with depressive symptoms and light business thoughts. The total prediction accuracy rate was 97.0%. The results are shown in Table 3.

We have planned a path analysis of the stress process (stressor → stress response) of college students that induces depressive symptoms and then produces light business thoughts. The results fit two structural equation models. The significance test found that the goodness of fit of each model was high (p<0.01). The results are shown in Table 4. Model 1 shows that the direct influence coefficient of stressor and stress response on light business is minimal (0.00, 0.09), so we can ignore it. It also shows that the direct induction coefficient (0.26) of the stressor on depressive symptoms is weaker than the direct induction coefficient (0.32) of the stress response on depressive symptoms [7]. The comparison between Model 1 and Model 2 shows that the influence coefficient of the stressor caused by stress response has increased from 0.80 of Model 1 to 0.83 in Model 2. The impact of the stress response on depressive symptoms increased from 0.32 in Model 1 to 0.56 in Model 2. The impact of depressive symptoms on light business thoughts also increased from 0.26 in Model 1 to 0.31 in Model 2. According to the reality that the stressor must pass the stress response to induce the depressive symptoms and possibly produce the objective path of lightness, we should choose Model 2 as the standard model for this study and Model 1 as the reference model. The model is shown in Figure 1.

Fig. 1

Structural equation Model 2 fits and produces corresponding analysis results of the direct and indirect effects of various factors in the stress process of college students on depressive symptoms and negligence in business [8]. Table 5 shows that the direct influence of the stressor on the stress response is 0.829. The indirect effect of the stress response on depressive symptoms is 0.465. The indirect effect of stress response and depressive symptoms on light business thoughts is 0.145. The direct impact of the stress response on depressive symptoms is 0.562. The indirect effect of depressive symptoms on light business ideology is 0.174. Finally, the effect of the depressive symptoms induced indirectly by the stressor and directly induced by the stress response on the light business idea is 0.311 (Table 5).