CN108052783A - A kind of unsaturated soil dynamic numerical calculation method based on adaptive step - Google Patents

Abstract

The invention discloses a numerical calculation method of unsaturated soil dynamics based on an adaptive step size, which solves the problems in the prior art that the calculation accuracy and efficiency cannot be well unified, and the time step size cannot be automatically adjusted, and has the ability to accurately Evaluate calculation errors and improve calculation efficiency, save calculation time, and can accurately describe the effect of deformation characteristics of unsaturated soil. The technical solution is: including the evaluation of solid phase displacement error, pore water pressure error and pore air pressure error. Calculate the solid phase displacement error, pore water pressure error, and pore air pressure error in the numerical analysis, use the mixed error as a function of the solid phase displacement error, pore water pressure error, and pore air pressure error, and use the mixed error as the basis for time step adjustment , accurately evaluate and calculate the numerical error of unsaturated soil dynamics, and adjust the calculation time step in real time.

Description

A Numerical Calculation Method of Unsaturated Soil Dynamics Based on Adaptive Step Size

technical field

The invention relates to the technical field of unsaturated soil mechanics research, in particular to an unsaturated soil dynamic numerical calculation method based on an adaptive step size.

Background technique

The study of unsaturated soil mechanics began in the 1930s. Unsaturated soil is characterized by three phases: solid, liquid and gas, and there is matrix suction at the liquid-gas interface. When calculating the stress in the soil The solid phase effective stress, pore water pressure and pore gas pressure need to be calculated separately.

In actual engineering, unsaturated soil accounts for the majority, for example, unsaturated soil is involved in foundation pit engineering, roadbed pavement engineering, and slope engineering. The research methods on unsaturated soil include theoretical analysis, experimental research and numerical analysis. As the third scientific research method, numerical analysis is favored by many researchers.

Numerical analysis is divided into static calculation and dynamic calculation. In finite element dynamic numerical calculation, it is necessary to discretize the control equations in space domain and time domain at the same time, and the discretization error cannot be ignored. The discretization error in the time domain is related to the time discretization method and the selection of the calculation time step. If the implicit solution method is used, higher accuracy can be obtained, but iterative calculations are required, which will have the disadvantages of non-convergence and low efficiency; The explicit solution method is efficient, but the selection of the calculation time step has a significant impact on the accuracy of the calculation results.

All in all, the following questions remain:

(1) In the explicit calculation, the accuracy and efficiency of the calculation cannot be well unified. The smaller the time step, the higher the accuracy and the longer the calculation time; the larger the time step, the shorter the calculation time, but the lower the accuracy. Not guaranteed;

(2) There is still a lack of a calculation method that can automatically adjust the time step in the numerical calculation of unsaturated soil dynamics.

Contents of the invention

In order to overcome the deficiencies of the prior art, the present invention provides a numerical calculation method for unsaturated soil dynamics based on an adaptive step size, which can accurately evaluate calculation errors and improve calculation efficiency, save calculation time, and accurately describe Effects on the deformation characteristics of unsaturated soils.

The present invention adopts following technical scheme:

A numerical calculation method of unsaturated soil dynamics based on adaptive step size, including the evaluation of solid phase displacement error, pore water pressure error and pore air pressure error, through real-time calculation of solid phase displacement error, pore water pressure error and pore pressure error in numerical analysis Pore air pressure error, using the mixed error as a function of solid phase displacement error, pore water pressure error and pore air pressure error, and using the mixed error as the basis for time step adjustment, accurately evaluates and calculates the error of unsaturated soil dynamics numerical calculation.

Further, the following steps are included:

Step (1) Set the control parameters, which are: initial time step Δt ₀ , allowable value of mixed error Solid phase displacement error adjustment coefficient λ _u , pore water pressure error adjustment coefficient λ _w , pore air pressure error adjustment coefficient λ _a , the minimum value f _min of the step adjustment coefficient and the maximum value f _max of the step adjustment coefficient;

Step (2) carries out the current step of finite element numerical calculation;

Step (3) performing solid phase displacement error evaluation, pore water pressure error evaluation and pore air pressure error evaluation;

Step (4) calculates mixing error, time step adjustment coefficient;

Step (5) Calculate the new time step and load item;

Step (6) compares the size of mixing error and error allowable value;

Among them, if the mixed error is less than or equal to the error allowable value, continue the numerical calculation of the next step, and adopt a new time step; if the mixed error is greater than the error allowable value, return to the current step for correction calculation, and adopt a new time step.

Further, in the step (3), the relative error of the pore pressure error is expressed as:

Among them, η ^a (t+Δt) is the relative error of pore pressure, is the maximum pore pressure.

Further, the pore pressure error e ^a (t+Δt) is expressed as:

in, is the second-order precision solution of pore pressure at time t+Δt, is the first-order precision solution of pore pressure at time t+Δt.

Further, the mixed error is expressed as:

Among them, η ^mix is mixed error, is the solid phase displacement error, is the pore water pressure error, is the pore pressure error.

Further, in the step (5), the new time step is expressed as:

Δt _new = f · Δt _old

Among them, Δt _new is the new step size after adjustment, Δt _old is the old step size before adjustment, and f is the step size adjustment coefficient.

Further, in the step (5), the adjustment formula of the load item is:

in, is the load value of the m+1th substep in the nth step, is the load value of the mth substep in the nth step, a _n+1 is the load value of the n+1th step, a _n is the load value of the nth step, Δt _max is the distance between the nth step and the n+1th step Time Step.

Compared with prior art, the beneficial effect of the present invention is:

(1) The present invention is based on the characteristics of unsaturated soil being composed of three phases of solid phase, liquid phase and gas phase, evaluating the solid phase displacement error, pore water pressure error and pore air pressure error, and obtaining unsaturated soil by evaluating the mixed error The time-domain discrete error of dynamic numerical calculation of three-phase medium can accurately describe the deformation characteristics of unsaturated soil;

(2) The present invention provides a time step correction formula, obtains a new time step, realizes the real-time adjustment of the load item, and corresponds to the time step, if the mixed error is greater than the allowable value of the error, the return correction can be realized, and a new time step can be carried out. calculation to achieve the purpose of improving calculation accuracy.

Description of drawings

The accompanying drawings constituting a part of the present application are used to provide further understanding of the present application, and the schematic embodiments and descriptions of the present application are used to explain the present application, and do not constitute improper limitations to the present application.

Fig. 1 is a flowchart of the present invention.

Detailed ways

It should be pointed out that the following detailed description is exemplary and intended to provide further explanation to the present application. Unless defined otherwise, all technical and scientific terms used herein have the same meaning as commonly understood by one of ordinary skill in the art to which this application belongs.

It should be noted that the terminology used here is only for describing specific implementations, and is not intended to limit the exemplary implementations according to the present application. As used herein, unless the context clearly dictates otherwise, the singular is intended to include the plural, and it should also be understood that when the terms "comprising" and/or "comprising" are used in this specification, they mean There are features, steps, operations, means, components and/or combinations thereof.

As introduced in the background technology, there are problems in the prior art that the calculation accuracy and efficiency cannot be well unified, and the time step cannot be automatically adjusted. In order to solve the above technical problems, this application proposes a method based on an adaptive step Numerical calculation method of unsaturated soil dynamics.

In a typical implementation of the present application, as shown in Figure 1, a numerical calculation method for unsaturated soil dynamics based on an adaptive step size is provided, including solid phase displacement error, pore water pressure error and pore air pressure error Evaluation, by calculating the solid phase displacement error, pore water pressure error, and pore air pressure error in the numerical analysis in real time, using the mixed error as a function of the solid phase displacement error, pore water pressure error, and pore air pressure error, and using the mixed error as a time step The basis for long adjustment, accurate evaluation and calculation of numerical errors of unsaturated soil dynamics.

The specific steps are:

1. Set the control parameters:

Define an initial time step Δt ₀ , the allowable value of the mixed error Solid phase displacement error adjustment coefficient λ _u , pore water pressure error adjustment coefficient λ _w , pore air pressure error adjustment coefficient λ _a , the minimum value f _min of the step size adjustment coefficient and the maximum value f _max of the step size adjustment coefficient; Calculate the current step.

2. Evaluation of solid phase displacement error, pore water pressure error evaluation and pore air pressure error evaluation:

The solid phase displacement error evaluation is based on the Newmark-β time domain discrete formula, and the acceleration error formula is obtained:

In formula (1), is the acceleration error, Δt is the calculation time step, is the acceleration value at time t, is the acceleration at time t+Δt, and τ is any value between t-t+Δt.

Integrating the acceleration error formula, the velocity error formula is obtained:

In formula (2) is speed error.

Integrate the velocity error formula to get the solid phase displacement error formula:

Equation (3) is the displacement error of e(t+Δt) solid phase.

Thus, the relative error formula of solid phase displacement is obtained:

In formula (4), η ^u (t+Δt) is the relative error of solid phase displacement, is the maximum value of the solid phase displacement.

The evaluation of the pore water pressure error is the difference between the central difference solution and the backward difference solution, which is a truncation error; the pore water pressure error is expressed as:

In formula (5), e ^w (t+Δt) is the pore water pressure error, is the rate of change of pore water pressure at time t, is the rate of change of pore pressure at time t+Δt.

The relative error of pore water pressure error is expressed as:

In formula (6), η ^w (t+Δt) is the relative error of pore water pressure, is the maximum pore water pressure.

The estimation of the pore pressure error is expressed as:

In formula (7), e ^a (t+Δt) is the pore pressure error, is the second-order precision solution of pore pressure at time t+Δt, is the first-order precision solution of pore pressure at time t+Δt.

The relative error of pore pressure is expressed as:

In formula (8), η ^a (t+Δt) is the relative error of pore pressure, is the maximum pore pressure.

3. Calculate the mixed error and time step adjustment coefficient:

Taking the mixed error as a function of solid phase displacement error, pore water pressure error and pore air pressure error, it is expressed as:

In formula (9), η ^mix is the mixing error, is the solid phase displacement error, Pore water pressure error, is the pore air pressure error, _λu is the adjustment coefficient of the solid phase displacement error, _λw is the pore water pressure error adjustment coefficient, and _λa is the pore air pressure error adjustment coefficient.

By setting three adjustment coefficients λ _u , λ _w and λ _a , the proportions of displacement error, pore water pressure error and pore air pressure error in the mixed error can be considered. Appropriate adjustments can be made for different working conditions. For example, for working conditions that are sensitive to pore water pressure, _λw should take a large value, and _λu and _λa should take smaller values; for working conditions where pore pressure and air pressure play a decisive role, Then λ _a takes a larger value, and λ _u and λ _w take a smaller value.

Among them, the value ranges of the three parameters are: _{0≤λu≤1 , 0≤λw≤1, 0≤λa≤1} .

The adjustment factor for the time step is expressed as:

In formula (10), f is the step size adjustment coefficient, Mixed error allowance.

4. Calculate the new time step and load item:

Limit the time step size in the range:

f _min ≤ f ≤ f _max (11)

In formula (11), f _min is the minimum value of the step size adjustment coefficient, and f _max is the maximum value of the step size adjustment coefficient.

The new time step is expressed as:

Δt _new = f · Δt _old (12)

In formula (12), Δt _new is the adjusted new step size, and Δt _old is the old step size before adjustment.

There is a one-to-one correspondence between the time step and the load value. After the time step is adjusted, the load item must be adjusted accordingly. Assuming that the load is a linear transformation, the load item is adjusted by the following formula:

In formula (13), is the load value of the m+1th substep in the nth step, is the load value of the mth substep in the nth step, a _n+1 is the load value of the n+1th step, a _n is the load value of the nth step, Δt _max is the distance between the nth step and the n+1th step Time Step.

5. Compare the size of the mixed error and the allowable error value; if the mixed error is less than or equal to the allowable error value, continue to the next step of calculation; if the mixed error is greater than the allowable error value, return to the previous step for correction calculation.

The entire calculation process is monitored in real time, and data such as solid phase displacement error, pore water pressure error, pore air pressure error, mixing error, step adjustment coefficient, and time step are output at each step.

The above descriptions are only preferred embodiments of the present application, and are not intended to limit the present application. For those skilled in the art, there may be various modifications and changes in the present application. Any modifications, equivalent replacements, improvements, etc. made within the spirit and principles of this application shall be included within the protection scope of this application.

Claims (7)

1. A method for calculating unsaturated soil dynamic numerical values based on self-adaptive step length is characterized by comprising the steps of evaluating solid phase displacement errors, pore water pressure errors and pore air pressure errors, calculating the solid phase displacement errors, the pore water pressure errors and the pore air pressure errors in numerical analysis in real time, taking mixed errors as functions of the solid phase displacement errors, the pore water pressure errors and the pore air pressure errors, taking the mixed errors as the basis of time step length adjustment, and accurately evaluating and calculating the errors of the unsaturated soil dynamic numerical value calculation.

2. The method for calculating the unsaturated soil dynamic value based on the adaptive step size according to claim 1, characterized by comprising the following steps of:

setting control parameters which are respectively as follows: initial time step Δ t₀Allowable value of mixing errorSolid phase displacement error adjustment coefficient lambda_uAdjustment coefficient lambda of pore water pressure error_wAdjustment coefficient lambda of pore pressure error_aMinimum value f of step adjustment coefficient_minAnd the maximum value f of the step-size adjustment coefficient_max；

Step (2) calculating the finite element numerical value in the current step;

step (3) solid phase displacement error evaluation, pore water pressure error evaluation and pore air pressure error evaluation are carried out;

calculating a mixing error and a time step length adjusting coefficient;

step (5) calculating a new time step and a new load item;

step (6) comparing the mixed error with the allowable error value;

if the mixing error is less than or equal to the error allowable value, continuing the next numerical calculation, and adopting a new time step; if the mixing error is larger than the error allowable value, returning to the current step for correction calculation, and adopting a new time step.

3. The method for calculating the unsaturated soil dynamic value based on the adaptive step size according to claim 2, wherein in the step (3), the relative error of the pore air pressure error is represented as:

<mrow> <msup> <mi>&amp;eta;</mi> <mi>a</mi> </msup> <mrow> <mo>(</mo> <mi>t</mi> <mo>+</mo> <mi>&amp;Delta;</mi> <mi>t</mi> <mo>)</mo> </mrow> <mo>=</mo> <mo>|</mo> <mfrac> <mrow> <msup> <mi>e</mi> <mi>a</mi> </msup> <mrow> <mo>(</mo> <mi>t</mi> <mo>+</mo> <mi>&amp;Delta;</mi> <mi>t</mi> <mo>)</mo> </mrow> </mrow> <msubsup> <mi>p</mi> <mrow> <mi>t</mi> <mo>+</mo> <mi>&amp;Delta;</mi> <mi>t</mi> </mrow> <mrow> <mi>a</mi> <mo>,</mo> <mi>max</mi> </mrow> </msubsup> </mfrac> <mo>|</mo> </mrow>

wherein, η^a(t + deltat) is the relative error in pore pressure,is the maximum value of the pore pressure.

4. The method for calculating the unsaturated soil dynamic value based on the adaptive step size as claimed in claim 3, wherein the pore air pressure error e^a(t + Δ t) is expressed as:

<mrow> <msup> <mi>e</mi> <mi>a</mi> </msup> <mrow> <mo>(</mo> <mi>t</mi> <mo>+</mo> <mi>&amp;Delta;</mi> <mi>t</mi> <mo>)</mo> </mrow> <mo>=</mo> <msubsup> <mi>p</mi> <mrow> <mi>t</mi> <mo>+</mo> <mi>&amp;Delta;</mi> <mi>t</mi> </mrow> <mrow> <mi>a</mi> <mn>2</mn> </mrow> </msubsup> <mo>-</mo> <msubsup> <mi>p</mi> <mrow> <mi>t</mi> <mo>+</mo> <mi>&amp;Delta;</mi> <mi>t</mi> </mrow> <mrow> <mi>a</mi> <mn>1</mn> </mrow> </msubsup> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>7</mn> <mo>)</mo> </mrow> </mrow>

wherein,a second order resolution of pore pressure at time t + deltat,is a first order accuracy solution of the pore pressure at time t + Δ t.

5. The method for calculating the unsaturated soil dynamic value based on the adaptive step size according to claim 2, wherein the mixing error is expressed as:

<mrow> <msup> <mi>&amp;eta;</mi> <mrow> <mi>m</mi> <mi>i</mi> <mi>x</mi> </mrow> </msup> <mo>=</mo> <msqrt> <mrow> <msub> <mi>&amp;lambda;</mi> <mi>u</mi> </msub> <msup> <mrow> <mo>(</mo> <msup> <mover> <mi>&amp;eta;</mi> <mo>&amp;OverBar;</mo> </mover> <mi>u</mi> </msup> <mo>)</mo> </mrow> <mn>2</mn> </msup> <mo>+</mo> <msub> <mi>&amp;lambda;</mi> <mi>w</mi> </msub> <msup> <mrow> <mo>(</mo> <msup> <mover> <mi>&amp;eta;</mi> <mo>&amp;OverBar;</mo> </mover> <mi>w</mi> </msup> <mo>)</mo> </mrow> <mn>2</mn> </msup> <mo>+</mo> <msub> <mi>&amp;lambda;</mi> <mi>a</mi> </msub> <msup> <mrow> <mo>(</mo> <msup> <mover> <mi>&amp;eta;</mi> <mo>&amp;OverBar;</mo> </mover> <mi>a</mi> </msup> <mo>)</mo> </mrow> <mn>2</mn> </msup> </mrow> </msqrt> </mrow>

wherein, η^mixIn order to mix the errors, the error rate of the mixing,in order to obtain a solid-phase displacement error,in order to determine the pore water pressure error,is the pore pressure error.

6. The method for calculating the unsaturated soil dynamic force value based on the adaptive step length as claimed in claim 2, wherein in the step (5), the new time step length is represented as:

Δt_new＝f·Δt_old

wherein, Δ t_newFor the adjusted new step size, Δ t_oldF is the step length adjusting coefficient before adjusting.

7. The method for calculating the unsaturated soil dynamic value based on the adaptive step size according to claim 2, wherein in the step (5), the adjustment formula of the load term is as follows:

<mrow> <msubsup> <mi>a</mi> <mi>n</mi> <mrow> <mi>m</mi> <mo>+</mo> <mn>1</mn> </mrow> </msubsup> <mo>=</mo> <msubsup> <mi>a</mi> <mi>n</mi> <mi>m</mi> </msubsup> <mo>+</mo> <mfrac> <mrow> <msub> <mi>&amp;Delta;t</mi> <mrow> <mi>n</mi> <mi>e</mi> <mi>w</mi> </mrow> </msub> </mrow> <mrow> <msub> <mi>&amp;Delta;t</mi> <mrow> <mi>m</mi> <mi>a</mi> <mi>x</mi> </mrow> </msub> </mrow> </mfrac> <mrow> <mo>(</mo> <msub> <mi>a</mi> <mrow> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>-</mo> <msub> <mi>a</mi> <mi>n</mi> </msub> <mo>)</mo> </mrow> </mrow>

wherein,is the load value of the (m + 1) th sub-step in the nth step,is the load value of the mth sub-step in the nth step, a_n+1Is the load value of step n +1, a_nIs the load value of step n, Δ t_maxIs the time step between the nth step and the (n + 1) th step.