Three-dimensional flexible distributed strain sensor and strain interpretation method for temperature compensation thereof

Abstract

The invention discloses a three-dimensional flexible distributed strain sensor and a strain interpretation method for temperature compensation thereof, and relates to the technical field of intelligent sensors. The three-dimensional flexible distributed strain sensor comprises a sensor body and a signal conditioning module; wherein the sensor body comprises an Ecoflex flexible substrate, a liquid metal microchannel network and an annular electrode array; the Ecoflex flexible substrate is internally provided with a three-dimensional communicated micro-channel structure formed by a three-dimensional printing sacrificial template process; the liquid metal micro-channel network is composed of gallium-indium alloy filled in the three-dimensional communicating micro-channel structure and comprises an X-direction micro-channel group extending in the X direction, a Y-direction micro-channel group extending in the Y direction and a Z-direction micro-channel group extending in the Z direction. The strain monitoring precision in a complex environment is remarkably improved, errors caused by temperature drift are effectively restrained, and the strain monitoring device has the advantages of being high in flexibility, high in stability and good in adaptability.

Description

Technical Field

[0001] This invention relates to the field of intelligent sensor technology, and in particular to a three-dimensional flexible distributed strain sensor and a strain interpretation method with temperature compensation. Background Technology

[0002] Three-dimensional flexible distributed strain monitoring technology has received increasing attention in recent years in the fields of civil engineering, transportation infrastructure, and soft structure health monitoring. As roadbeds, large-volume flexible structures, and heterogeneous materials inevitably undergo multi-axial coupled deformation during long-term service, traditional strain sensing technologies based on single-axis or two-dimensional planar measurement methods struggle to simultaneously capture the overall deformation characteristics in multiple directions—longitudinal, transverse, and vertical—especially in real-time identification of roadbed settlement angles, shear deformation, and localized stress concentration areas. Therefore, researching flexible strain sensors that can deform along with the structure and generate a continuous distributed response in three-dimensional space has become an important direction for technological development.

[0003] Most existing flexible strain sensors employ metal thin films, metal nanowires, graphene composites, or carbon nanotube composites as sensing elements, calculating strain states through changes in resistance. These material structures typically exhibit two-dimensional characteristics: in-plane continuity and limited thickness. While their sensing elements are highly sensitive to tension or compression along the film surface, their response to changes in thickness is limited, making it difficult to obtain independent and reliable X, Y, and Z strain components in three-dimensional space. To extend three-dimensional sensing capabilities to some extent, some technical solutions attempt to achieve multi-directional response by stacking multiple thin films or arranging conductive traces in different directions. However, the interlayer bonding strength of these structures is limited, making them prone to delamination, fatigue damage, or localized circuit breakage under long-term alternating loads, thus affecting the continuity and stability of measurements. Summary of the Invention

[0004] The purpose of this invention is to provide a three-dimensional flexible distributed strain sensor and its temperature-compensated strain interpretation method, which has the advantages of high flexibility, strong stability and good adaptability.

[0005] To solve the above-mentioned technical problems, the technical solution adopted by the present invention is as follows: On one hand, a three-dimensional flexible distributed strain sensor is provided, including a sensor body and a signal conditioning module. The sensor body comprises an Ecoflex flexible substrate, a liquid metal microchannel network, and a ring electrode array. The Ecoflex flexible substrate contains a three-dimensional interconnected microchannel structure formed using a three-dimensional printing sacrificial template process. The liquid metal microchannel network is composed of a gallium-indium alloy filled within the three-dimensional interconnected microchannel structure. The liquid metal microchannel network includes an X-direction microchannel group extending along the X direction, a Y-direction microchannel group extending along the Y direction, and a Z-direction microchannel group extending along the Z direction. The X, Y, and Z directions are orthogonal to each other. The channel group, Y-axis microchannel group, and Z-axis microchannel group are interconnected at their spatial intersection; the ring electrode array is made of a flexible printed circuit board and includes multiple electrode units uniformly distributed along the circumference of the Ecoflex flexible substrate, with each electrode unit electrically connected to a corresponding port of the liquid metal microchannel network; the signal conditioning module includes a direct digital frequency synthesis signal source, a current injection switching matrix, and a lock-in amplifier detector; the direct digital frequency synthesis signal source is used to generate a sweep frequency excitation signal; the current injection switching matrix is ​​connected to all electrode units of the ring electrode array; the lock-in amplifier detector is used to detect the voltage response signal and extract amplitude and phase information.

[0006] Furthermore, the ring electrode array comprises thirty-two electrode units.

[0007] Furthermore, the sweep excitation signal generated by the direct digital frequency synthesis signal source covers the low-frequency to high-frequency bands.

[0008] Furthermore, the current injection switching matrix is ​​used to switch the combination of excitation electrode pairs and measurement electrode pairs according to the pattern of injecting current into adjacent electrodes and measuring voltage on the opposite electrode.

[0009] On the other hand, a strain interpretation method for temperature compensation of a three-dimensional flexible distributed strain sensor is provided, including the following steps: Step 1: Generate a sweep excitation signal using a direct digital frequency synthesis signal source, and complete the measurement by sequentially traversing all electrode combinations using a current injection switching matrix. Collect the voltage response amplitude and voltage response phase using a lock-in amplifier detector and calculate the complex impedance measurement value to obtain the original dataset of the multi-frequency impedance spectrum. Step 2: Perform frequency-domain alternating iterative resistance tomography reconstruction processing based on total variation constraints on the original multi-frequency impedance spectrum dataset to output low-frequency resistivity field distribution images and high-frequency resistivity field distribution images; Step 3: Calculate the temperature field distribution data and the temperature-compensated resistivity field distribution data based on the low-frequency resistivity field distribution image and the high-frequency resistivity field distribution image; Step 4: Calculate and output the X-axis strain field distribution, Y-axis strain field distribution, and Z-axis strain field distribution based on the temperature-compensated resistivity field distribution data.

[0010] Furthermore, in step one, the frequency sweep excitation signal gradually increases from the starting frequency to the ending frequency. After completing the measurement of all electrode combinations at each excitation frequency, all complex impedance measurement values ​​corresponding to the current excitation frequency are stored as a frame of frequency impedance data. The frequency impedance data of all excitation frequencies constitute the original dataset of multi-frequency impedance spectrum.

[0011] Furthermore, in step two, the frequency domain alternating iterative resistance tomography reconstruction process includes: establishing a three-dimensional hexahedral mesh model for the monitoring area of ​​the sensor body, dividing it into multiple hexahedral mesh units at equal intervals along the X, Y, and Z directions, and assigning a three-dimensional mesh index to each hexahedral mesh unit; establishing a dual-channel state register for each hexahedral mesh unit, the dual-channel state register including a first channel for storing low-frequency resistivity estimates and a second channel for storing high-frequency resistivity estimates; extracting low-frequency impedance data frames and high-frequency impedance data frames from the original multi-frequency impedance spectrum dataset; and entering the frequency domain alternating iterative main loop, each round of the frequency domain alternating iterative main loop including a low-frequency channel update stage, a high-frequency channel update stage, and a total variational smoothing stage executed sequentially.

[0012] Furthermore, the execution process of the low-frequency channel update stage is as follows: Based on the low-frequency resistivity estimates of all hexahedral mesh elements, a finite element forward electric field solution is performed to obtain the theoretical low-frequency impedance vector; the theoretical low-frequency impedance vector is subtracted element-by-element from the corresponding measured low-frequency impedance vector in the low-frequency impedance data frame to obtain the low-frequency impedance residual vector; a back projection operation is performed on each residual element in the low-frequency impedance residual vector to determine the set of main influencing elements corresponding to each residual element; the value of each residual element is evenly distributed to each hexahedral mesh element in the corresponding set of main influencing elements as a low-frequency resistivity correction increment; for each hexahedral mesh element, all accumulated low-frequency resistivity correction increments are summed to obtain the total low-frequency resistivity correction amount; the total low-frequency resistivity correction amount is added to the current low-frequency resistivity estimate of the hexahedral mesh element and written back to the first channel; the high-frequency channel update stage uses the same method as the low-frequency channel update stage to update the high-frequency resistivity estimates in the second channel.

[0013] Furthermore, the execution process of the total variation smoothing stage is as follows: For each hexahedral mesh element, calculate the low-frequency total variation gradient value. The low-frequency total variation gradient value is the sum of the absolute values ​​of the low-frequency resistivity differences in the X, Y, and Z directions of the current hexahedral mesh element. Based on the low-frequency total variation gradient value, divide all hexahedral mesh elements into low-frequency high-gradient elements and low-frequency flat elements. For each low-frequency flat element, perform a low-frequency neighborhood mean smoothing operation, taking the arithmetic mean of the low-frequency resistivity estimates of all neighboring elements of the current low-frequency flat element and the currently stored low-frequency resistivity estimate of the current low-frequency flat element, and then writing it back to the first channel. The same smoothing operation is performed on the high-frequency resistivity estimates in the second channel.

[0014] Furthermore, in step three, the following operations are performed on each hexahedral mesh cell: the low-frequency resistivity estimate of the current hexahedral mesh cell is subtracted from the high-frequency resistivity estimate of the current hexahedral mesh cell to obtain the frequency domain resistivity difference; the frequency domain resistivity difference is divided by the unit temperature resistivity change coefficient to obtain the temperature deviation estimate; the temperature deviation estimate is added to the ambient reference temperature to obtain the absolute temperature estimate; and the product of the temperature deviation estimate and the unit temperature resistivity change coefficient is subtracted from the high-frequency resistivity estimate of the current hexahedral mesh cell to obtain the temperature-compensated resistivity value.

[0015] The three-dimensional flexible distributed strain sensor and its temperature-compensated strain interpretation method of the present invention have the following beneficial effects: The present invention constructs a three-dimensional interconnected microchannel structure within a flexible substrate and fills it with liquid metal to form a liquid metal microchannel network. This allows the sensor to freely deform in three-dimensional space along with the structure, generating impedance changes highly consistent with the actual stress state, thereby achieving synchronous monitoring of the X-axis, Y-axis, and Z-axis strain components. Since the microchannels are orthogonal to each other in the X, Y, and Z directions and interconnected at their spatial intersections, the present invention can obtain truly three-dimensional distributed strain information within a single flexible body, avoiding the limitation of traditional two-dimensional sensitive materials that can only acquire in-plane strain.

[0016] Meanwhile, this invention employs a combination of a ring electrode array and multi-frequency sweep excitation, enabling each electrode combination to capture the differentiated responses of the liquid metal microchannel network in terms of interface polarization, geometric changes, and temperature perturbations at different excitation frequencies. This results in a rich dataset of original multi-frequency impedance spectra, significantly increasing the amount of information compared to traditional single-frequency measurement methods. Furthermore, this invention introduces a frequency-domain alternating iterative resistance tomography reconstruction method. By distinguishing the resistivity field distribution under low-frequency and high-frequency conditions, it allows temperature-induced changes to be separated from geometrically strain-induced changes. Combined with total variational smoothing constraints, the reconstruction results are more stable and continuous, effectively suppressing noise propagation and local non-physical processes.

[0017] Furthermore, by constructing temperature field distribution data and performing temperature compensation, the resistivity field distribution data after temperature compensation reflects only the strain effect, ultimately calculating the strain field distribution in three dimensions and improving interpretation accuracy. This invention features high overall structural flexibility, strong resolution, and good reliability, enabling long-term operation in complex environments. Compared to existing technologies, this invention achieves true distributed strain measurement in three-dimensional space and effectively solves the measurement error problem caused by temperature drift, showing promising application prospects in large-scale civil engineering and flexible structure monitoring. Attached Figure Description

[0018] Figure 1 A top view of the ring electrode array configuration and current injection measurement mode of the three-dimensional flexible distributed strain sensor provided in an embodiment of the present invention; Figure 2 This is a schematic diagram showing the comparison results of temperature field distribution, relative change of resistivity before compensation, and relative change of resistivity after compensation in the temperature compensation strain interpretation method provided in the embodiments of the present invention. Figure 3 A schematic diagram of the component distribution of the three-dimensional strain field obtained after temperature compensation processing in the X, Y and Z directions provided in an embodiment of the present invention; Figure 4 This is a schematic diagram showing the comparison results of low-frequency resistivity field distribution images and high-frequency resistivity field distribution images obtained by frequency-domain alternating iterative resistivity tomography reconstruction processing based on total variation constraints, as provided in an embodiment of the present invention. Detailed Implementation

[0019] Example 1: A three-dimensional flexible distributed strain sensor, comprising a sensor body and a signal conditioning module; wherein, the sensor body includes an Ecoflex flexible substrate, a liquid metal microchannel network, and a ring electrode array; the Ecoflex flexible substrate contains a three-dimensional interconnected microchannel structure formed by a three-dimensional printing sacrificial template process; the liquid metal microchannel network is composed of a gallium-indium alloy filled within the three-dimensional interconnected microchannel structure, and the liquid metal microchannel network includes an X-direction microchannel group extending along the X direction, a Y-direction microchannel group extending along the Y direction, and a Z-direction microchannel group extending along the Z direction, the X, Y, and Z directions being orthogonal to each other, and the X-direction microchannel... The channel group, Y-axis microchannel group, and Z-axis microchannel group are interconnected at their spatial intersection; the ring electrode array is made of a flexible printed circuit board and includes multiple electrode units uniformly distributed along the circumference of the Ecoflex flexible substrate, with each electrode unit electrically connected to a corresponding port of the liquid metal microchannel network; the signal conditioning module includes a direct digital frequency synthesis signal source, a current injection switching matrix, and a lock-in amplifier detector; the direct digital frequency synthesis signal source is used to generate a sweep frequency excitation signal; the current injection switching matrix is ​​connected to all electrode units of the ring electrode array; the lock-in amplifier detector is used to detect the voltage response signal and extract amplitude and phase information.

[0020] The implementation of a three-dimensional flexible distributed strain sensor can be approached by considering the overall structure, the bottom-up manufacturing process, and the electrical excitation and measurement links. The following section presents a relatively complete implementation method as the main thread, while also providing alternative implementation methods at key points.

[0021] In its implementation, the Ecoflex flexible matrix is ​​first formed. A commercially available two-component silicone rubber material can be used as the Ecoflex flexible matrix material, with a Shore hardness of approximately [value missing]. arrive The relatively low Shore hardness allows the Ecoflex flexible matrix to undergo significant deformation under tension, compression, and bending, thereby inducing substantial geometric changes in the internal liquid metal microchannel network. The Ecoflex flexible matrix can be manufactured by mixing component A and component B in a specific mass ratio... After mixing, thorough stirring, and degassing in a vacuum environment, the mixture is poured into a mold for shaping. To form a three-dimensional interconnected microchannel structure within the Ecoflex flexible matrix, a three-dimensional interconnected sacrificial template prepared by a three-dimensional printing sacrificial template process is placed inside the mold before pouring the material.

[0022] Three-dimensional connected sacrificial templates can be formed in one step using water-soluble photosensitive resin or polyvinyl alcohol materials through 3D printing equipment. For example, they can be formed in the X, Y, and Z directions at intervals of approximately... Multiple straight channel frames are arranged, and the cross-sectional width and height of the channel frames can be... arrive Within a selected range, the channel framework in the X direction is designed with interconnected small cubic connecting cavities at the intersections with the channel frameworks in the Y and Z directions. This ensures the connectivity of the subsequently formed three-dimensional interconnected microchannel structure in the X, Y, and Z directions. After the Ecoflex flexible substrate has cured, the entire cured body is immersed in warm water or an alkaline solution to dissolve the sacrificial template material and drain it through pre-made drainage holes. This preserves a three-dimensional interconnected microchannel structure with a shape consistent with the sacrificial template within the Ecoflex flexible substrate.

[0023] The resulting three-dimensional interconnected microchannel structure is naturally divided into X-direction microchannel groups extending along the X direction, Y-direction microchannel groups extending along the Y direction, and Z-direction microchannel groups extending along the Z direction. The X, Y, and Z directions are orthogonal to each other, and the X, Y, and Z-direction microchannel groups are interconnected at their spatial intersections. This can be viewed as constructing a cavity network with a regular three-dimensional mesh shape within the Ecoflex flexible substrate. In this structure, when the Ecoflex flexible substrate is subjected to external three-dimensional strain, the cross-section and length of the microchannels in different directions change to varying degrees, thus ensuring that the liquid metal microchannel network has considerable sensitivity to multiaxial strain.

[0024] In another alternative implementation, the cross-section of the three-dimensional connected microchannel structure can be designed to be approximately circular to reduce local stress concentration. In this case, the sacrificial template can be printed using a near-cylindrical filament structure with a diameter of... The balance between spatial resolution and flow resistance is controlled by changing the longitudinal spacing between the X-axis microchannel group, Y-axis microchannel group, and Z-axis microchannel group.

[0025] After the three-dimensional interconnected microchannel structure is formed, a liquid metal microchannel network is created by filling the interior of the structure with a gallium-indium alloy. The gallium-indium alloy can be selected from approximately [weight percentage missing]. Gallium and The indium content is carefully controlled to maintain a liquid state at room temperature while possessing conductivity close to that of metallic wires. To achieve uniform filling, several injection and vent ports are created at the edge of the Ecoflex flexible substrate. A gallium-indium alloy is slowly injected through a syringe at one end, while a vacuum pump is connected to the other end to evacuate the gas. Under negative pressure, the gallium-indium alloy gradually fills all X-axis, Y-axis, and Z-axis microchannel groups along the three-dimensional interconnected microchannel structure until it overflows from the vent ports. The injection and vent ports are then sealed, completing the formation of the liquid metal microchannel network.

[0026] In liquid metal microchannel networks, local resistance is closely related to channel geometry. For a given segment of a microchannel, resistance can be used as a metric. Describe, in which This indicates the resistance corresponding to that microchannel segment. This indicates the resistivity of gallium-indium alloys. This indicates the effective length of the microchannel segment. This represents the effective cross-sectional area of ​​the microchannel segment. When the Ecoflex flexible substrate is stretched in the X direction, the area of ​​the microchannel group in the X direction... Enlarge Decrease, corresponding The resistance increases accordingly; the opposite occurs when compressed in the X direction. Microchannel arrays in the Y and Z directions also exhibit similar geometric changes under different strain states. Through the three-dimensional interconnected morphology, these local resistance changes are superimposed on the entire liquid metal microchannel network, allowing strain in different directions to be expressed through overall impedance changes.

[0027] Besides single-component gallium-indium alloys, small amounts of surfactants can be added to the gallium-indium alloy to improve the wetting effect between the liquid metal and the inner wall of the three-dimensional interconnected microchannel structure, thereby reducing the interfacial resistance caused by droplet splitting. In some implementations, trace amounts of conductive particles can also be mixed into the gallium-indium alloy to adjust the temperature coefficient, making the liquid metal microchannel network respond more linearly to temperature changes, which facilitates subsequent temperature compensation.

[0028] The ring electrode array is fabricated from a flexible printed circuit board for reliable electrical connection to a liquid metal microchannel network. The flexible printed circuit board can use polyimide as a substrate, with a thickness of approximately [missing information]. Forming a thickness of approximately [thickness missing] on the substrate The copper wire pattern is formed, and nickel and gold plating are applied to the electrode surface. A ring electrode array with multiple electrode units evenly distributed circumferentially along the Ecoflex flexible substrate is implemented. In one embodiment, the ring electrode array includes thirty-two electrode units, with an angular spacing of approximately [missing information]. Each electrode unit can be designed with a diameter of approximately The circular electrode pads are connected to the connection pads at one end of the flexible printed circuit board via serpentine traces. The serpentine traces can improve the mechanical compliance of the flexible printed circuit board during bending and reduce the constraints on the Ecoflex flexible substrate and liquid metal microchannel network.

[0029] refer to Figure 1This figure clearly illustrates the spatial configuration of 32 electrode units evenly distributed along the circumference of the sensor cross-section and the working principle of measurement on opposite sides of adjacent excitations. In this embodiment, the sensor body has a circular cross-section with a diameter of approximately 8 mm. The 32 electrode units of the ring electrode array are evenly distributed along the circumference, with an angular interval of 11.25 degrees between adjacent electrode units. Each electrode unit is connected to the signal conditioning module via conductive traces on a flexible printed circuit board, forming a complete electrical measurement path. In the figure, the light gray circular area represents the cross-section of the sensor body, the black solid circle marks the outer boundary of the sensor, and the internal liquid metal microchannel network constitutes a three-dimensional conductive path. Among the 32 evenly distributed electrode units, the small gray dots represent the electrode positions under normal conditions, providing complete boundary measurement capabilities.

[0030] Electrodes 1 and 2 are highlighted with large red circles in the diagram. These two adjacent electrode units constitute the excitation electrode pair in the current measurement configuration. A current injection switching matrix connects them to the output and return terminals of the constant current source to inject AC excitation current into the liquid metal microchannel network. Corresponding to the excitation electrode pair, electrodes 17 and 18, located on the opposite side of the sensor cross-section, are marked with blue squares. They constitute the measurement electrode pair and are connected to the differential voltage input of the lock-in amplifier detector via the current injection switching matrix to acquire the boundary voltage response signal generated by the current distribution. Thin green lines schematically depict the main conduction paths of current flowing from the excitation electrode pair to the measurement electrode pair. These paths exhibit a distribution characteristic of diffusion from the excitation electrode region towards the sensor center and then converging back to the measurement electrode region, reflecting the complex three-dimensional current distribution pattern within the liquid metal microchannel network. Since the liquid metal microchannel network has conductive paths in the X, Y, and Z directions, current can propagate along multiple spatial paths. The resistivity differences along different paths are comprehensively reflected in the voltage amplitude and phase information acquired by the measurement electrode pair.

[0031] During actual measurement, the current injection switching matrix sequentially switches the combination of excitation electrode pairs and measurement electrode pairs according to a predetermined measurement sequence. For example, in the first measurement, electrodes 1 and 2 are selected as the excitation electrode pair and electrodes 17 and 18 as the measurement electrode pair; in the second measurement, electrodes 2 and 3 are selected as the excitation electrode pair and electrodes 18 and 19 as the measurement electrode pair, and so on, completing the traversal of all electrode combinations. This adjacent excitation and opposite-side measurement mode ensures that current is injected from the boundary local region and voltage is measured in the opposite region, making each measurement highly sensitive to changes in resistivity distribution in a specific region inside the sensor. By systematically changing the positions of the excitation electrode pairs and measurement electrode pairs, the distribution characteristics of the internal three-dimensional resistivity field can be perceived from different angles and paths. The large amount of boundary measurement data obtained provides sufficient information redundancy for subsequent three-dimensional resistivity tomography reconstruction. The coordinate system is also marked in the figure, with the X direction horizontal to the right and the Y direction vertically upward, which is consistent with the physical coordinate system of the sensor body, facilitating the mapping of measurement data to a three-dimensional spatial grid model for resistivity inversion calculation. The electrode configuration and measurement mode design are the key technological foundation for realizing high spatial resolution three-dimensional strain field reconstruction. Through dense boundary electrical measurements, the changes in conductivity of liquid metal microchannel networks under different strain and temperature conditions can be captured, providing reliable raw data input for temperature-compensated strain interpretation.

[0032] To achieve electrical connection between the ring electrode array and the liquid metal microchannel network, through-holes corresponding to the electrode unit positions can be created circumferentially on the outer side of the Ecoflex flexible substrate, exposing the ports of the three-dimensional interconnected microchannel structure at the bottom of the through-holes. When the flexible printed circuit board is bonded to the outer surface of the Ecoflex flexible substrate, the electrode pads of each electrode unit are aligned with the corresponding through-hole positions, and conductive adhesive or conductive silver paste is filled into the through-holes, allowing the electrode pads to directly contact the gallium-indium alloy, thereby forming a stable connection with low contact resistance. Using conductive adhesive, reliable connections are maintained even under mechanical vibration and repeated bending conditions. Furthermore, the conductive adhesive itself has a certain degree of flexibility, which can buffer the relative displacement between the Ecoflex flexible substrate and the flexible printed circuit board.

[0033] In some alternative implementations, to improve the interfacial stability between the electrode and the liquid metal, a roughening structure or metalophilic coating can be added to the inner wall of the via and the electrode pads. This allows the gallium-indium alloy to form a stable adhesion morphology at the interface, thereby reducing the impact of changes in the interfacial contact area on the measurement results. Alternatively, depending on the installation space, the ring electrode array can be designed as a multi-ring structure, i.e., two or three rings of electrode units are arranged axially on the Ecoflex flexible substrate to obtain richer three-dimensional boundary measurement data.

[0034] The direct digital frequency synthesis signal source in the signal conditioning module is used to generate a frequency sweep excitation signal. In practice, a digital structure with a phase accumulator and waveform lookup table can be used, with a stable clock as a reference, to generate a frequency-adjustable sine wave output. For example, the reference clock frequency can be selected as... By changing the phase accumulation step value, the center frequency of the output excitation signal is adjusted to... arrive Within the range, the frequency step resolution can reach The output sine wave is fed into the current excitation circuit after digital-to-analog conversion and low-pass filtering, and is finally injected into the selected excitation electrode pair in the form of a constant current.

[0035] In the multi-frequency excitation mode, the frequency sweep excitation signal gradually increases from the starting frequency to the ending frequency. In the low-frequency range, the charge polarization effect at the interfaces of the liquid metal microchannel network, the Ecoflex flexible substrate, and the electrodes is more pronounced, with equivalent resistance and equivalent capacitance jointly affecting the complex impedance. In the high-frequency range, the influence of interface polarization is relatively weakened, and the measurement more prominently highlights the resistive characteristics of the liquid metal bulk. By continuously measuring multiple frequency points, a raw dataset of multi-frequency impedance spectra can be generated, providing a foundation for subsequent differentiation between temperature changes and strain changes based on frequency domain information.

[0036] The current injection switching matrix is ​​connected to all electrode units of the ring electrode array, responsible for switching the combination of excitation electrode pairs and measurement electrode pairs among the thirty-two electrode units according to a predetermined measurement mode. In one implementation, the current injection switching matrix can be constructed using a multi-channel analog switch array, with each electrode unit corresponding to one analog switch channel. Control logic connects two selected electrode channels to the current source output and current source return terminals as excitation electrode pairs, and connects the other two electrode channels to the differential voltage measurement input terminals as measurement electrode pairs. The measurement mode can be set to inject current into adjacent electrodes and measure voltage at opposite electrodes. For example, in the thirty-two electrode units, the numbered... and The electrode units are configured as excitation electrode pairs, and the numbered ones are... and The electrode unit is configured as a measuring electrode pair, wherein Indicates the current starting electrode number, in arrive The values ​​are retrieved in a loop.

[0037] The injection current from adjacent electrodes can form a locally concentrated current injection region on the annular boundary, allowing the current to enter the liquid metal microchannel network primarily from the space between the two excitation electrode units. Meanwhile, the voltage measured by the opposite electrode can geometrically span the entire cross-section, thus exhibiting strong sensitivity to changes in the internal resistivity distribution. With... As the excitation electrode pair increases, it moves gradually along the circumference, and the measurement electrode pair on the opposite side moves synchronously, forming a large number of measurement configurations covering the entire annular boundary. The measurement set obtained in this way meets the measurement requirements of three-dimensional resistivity tomography reconstruction and is beneficial for recovering the resistivity field distribution inside the liquid metal microchannel network in subsequent steps.

[0038] A lock-in amplifier (LPA) detector is used to detect voltage response signals and extract amplitude and phase information. In its implementation, the LPA receives a differential voltage signal acquired between the measuring electrode pairs and simultaneously obtains a reference signal of the same frequency and phase from a direct digital frequency synthesis (DFD) signal source. The LPA can synchronously detect the measured voltage signal and the reference signal, decomposing the measured voltage signal into in-phase and quadrature components. For example, the in-phase component can be represented as... Represent the orthogonal components as ,in This represents the voltage component that is in phase with the reference signal. Indicates the phase difference with the reference signal The voltage components. The voltage amplitude can be calculated from these two components. and voltage phase ,in It can be done get, It can be done Received, among which This represents the phase angle of the measured voltage relative to the reference signal.

[0039] Given the amplitude of the injected current and voltage amplitude In this case, the complex impedance amplitude under the corresponding measurement configuration can be obtained. Their relationship is ,in Indicates the magnitude of the impedance. This indicates the measured voltage amplitude. This represents the amplitude of the injected current. If the phase of the current is also measured, the phase of the complex impedance can be further obtained, thus providing a complete description of the complex impedance value of the measurement path. For each excitation frequency and each electrode combination, the lock-in amplifier detector outputs the corresponding amplitude and phase information. After traversing all electrode combinations at all excitation frequencies, a raw dataset of multi-frequency impedance spectra can be formed.

[0040] Example 2: A strain interpretation method for temperature compensation using a three-dimensional flexible distributed strain sensor, comprising the following steps: Step 1: Generate a sweep excitation signal using a direct digital frequency synthesis signal source, and complete the measurement by sequentially traversing all electrode combinations using a current injection switching matrix. Collect the voltage response amplitude and voltage response phase using a lock-in amplifier detector and calculate the complex impedance measurement value to obtain the original dataset of the multi-frequency impedance spectrum. Step 2: Perform frequency-domain alternating iterative resistance tomography reconstruction processing based on total variation constraints on the original multi-frequency impedance spectrum dataset to output low-frequency resistivity field distribution images and high-frequency resistivity field distribution images; Step 3: Calculate the temperature field distribution data and the temperature-compensated resistivity field distribution data based on the low-frequency resistivity field distribution image and the high-frequency resistivity field distribution image; Step 4: Calculate and output the X-axis strain field distribution, Y-axis strain field distribution, and Z-axis strain field distribution based on the temperature-compensated resistivity field distribution data.

[0041] In step one, the frequency sequence of the sweep excitation signal is first set using a direct digital frequency synthesis signal source. A set of discrete excitation frequencies can be pre-configured in the control program, and this set of excitation frequencies is represented as... ,in Indicates the first One excitation frequency, The index indicating the frequency. This indicates the total number of frequencies used in this frequency sweep. The unit is Hertz. In practical applications, you can choose... equal ,let near hertz, near The frequency band is set at Hertz, while other frequencies are distributed uniformly on the order of magnitude. This arrangement is because the low-frequency band is more sensitive to slow changes caused by interfacial polarization and temperature, while the high-frequency band better reflects the differences in conductive paths corresponding to geometric changes in the liquid metal bulk and the three-dimensional interconnected microchannel structure. By covering a wide frequency range, temperature-related and strain-related components can be more clearly distinguished in subsequent processing.

[0042] At a certain frequency Below, the direct digital frequency synthesis signal source outputs a sinusoidal excitation voltage with stable amplitude and phase, which is then limited by a constant current excitation circuit to produce an AC excitation current with a constant amplitude. This AC excitation current can be represented by a complex number as... ,in This represents the complex form of the alternating current injected into the excitation electrode pair at the current frequency, containing both amplitude and phase information. (This is incorrect.) Instead of abbreviating it, it is directly regarded as a complex current quantity containing two parameters: amplitude and phase.

[0043] At this frequency, the current injection switching matrix sequentially selects the excitation electrode pairs and measurement electrode pairs according to a preset measurement order. For example, taking thirty-two electrode units as an example, it can follow a mode of injecting current into adjacent electrodes and measuring voltage on opposite electrodes, starting from the... The first electrode unit and the first Starting with the first electrode unit as the first set of excitation electrode pairs, the corresponding measurement electrode pair can be selected from the first set. The first electrode unit and the first The first electrode unit. After completing a set of measurements, the excitation electrode pair moves sequentially to the second electrode unit. The first electrode unit and the first The electrode unit, the corresponding movement of the measuring electrode pair is the first electrode unit. The first electrode unit and the first Each electrode unit is individually selected, and so on, until all excitation and measurement electrode pairs have been traversed. The injected current from adjacent electrodes has the characteristic of causing a large amount of current to enter the liquid metal microchannel network from the local boundary region, while the measured voltage from the opposite electrode geometrically spans the entire cross-section. The combination of these two factors allows a single measurement to have strong sensitivity to the internal three-dimensional resistivity distribution. As the combination moves continuously, the current distribution and voltage sampling positions on the entire annular boundary undergo systematic changes, forming a large number of measurement paths suitable for three-dimensional resistance tomography reconstruction.

[0044] In each combination of excitation electrode pair and measurement electrode pair, the lock-in amplifier detector receives the differential voltage signal between the measurement electrode pairs and simultaneously obtains a reference signal of the same frequency from a direct digital frequency synthesis signal source. Internally, the lock-in amplifier detector performs synchronous coherent detection on the measurement voltage signal and the reference signal, decomposing the measurement voltage into in-phase and quadrature components. The in-phase component can be denoted as... The orthogonal components are denoted as ,in This represents the voltage component that is in phase with the reference signal. This represents the voltage component that is 90 degrees out of phase with the reference signal. These two components are used to construct the voltage amplitude and phase. It can be done through formula The calculation yielded, where The scalar magnitude of voltage is represented by the symbol. Represents the square root operation, exponent This indicates squaring the corresponding variable. Voltage phase. It can be done through formula The calculation yielded, where Represents the voltage phase angle, function Represents the arctangent function, numerator Represents orthogonal components, denominator This represents the in-phase component. These two expressions allow us to simultaneously obtain the frequency of the current measurement electrode pair. The voltage amplitude and voltage phase.

[0045] Given the complex form of the current The complex form of voltage Then, the corresponding complex impedance measurement value can be calculated. Their relationship is ,in This indicates the current combination of the excitation electrode pair and the measurement electrode pair at a frequency. The complex impedance measurement value below, Indicates the voltage amplitude and voltage phase The complex voltage formed by recombination, denominator This represents a complex current. Using this formula, the amplitude and phase information can be uniformly represented as a complex impedance, which can be used for subsequent resistance tomography reconstruction calculations.

[0046] In frequency Next, the current injection switching matrix traverses all electrode combinations, and the lock-in amplifier detector measures the complex impedance values ​​of each excitation electrode pair and measurement electrode pair. By storing these complex impedance measurements sequentially into a data structure, this set of complex impedance measurements can be considered as a frame of frequency impedance data. Let this frame contain... For each measurement configuration, this frame of data can be represented by a vector, denoted as . ,in Indicates frequency The complex impedance measurement vector obtained below, Indicates frequency Next The complex impedance measurement of each electrode assembly Indicates the measurement sequence number. Indicates the number of measurement configurations at this frequency, superscript. This represents the vector transpose operation. With... from Increase to ,all This constitutes the original dataset of multi-frequency impedance spectroscopy. This dataset simultaneously covers the response characteristics of three-dimensional liquid metal microchannel networks in both the frequency dimension and the spatial measurement path dimension, enabling subsequent processing to utilize both the low-frequency sensitivity to temperature and interface and the high-frequency sensitivity to geometric deformation.

[0047] In step two, before performing frequency-domain alternating iterative resistance tomography reconstruction processing based on total variational constraints on the original multi-frequency impedance spectrum dataset, a three-dimensional hexahedral mesh model corresponding to the three-dimensional flexible distributed strain sensor needs to be constructed first. The monitoring area can be divided into equal intervals in the X, Y, and Z directions, for example, divided in the X direction as follows: Layers, divided in the Y direction Layers, divided in the Z direction Layers. Each small volumetric unit is treated as a regular hexahedral mesh unit and assigned a 3D mesh index, denoted as . ,in This indicates the layer number of the hexahedral mesh element in the X direction. This indicates the layer number in the Y direction. This indicates the layer number in the Z direction. This indexing method matches the X-axis, Y-axis, and Z-axis microchannel groups within a three-dimensional interconnected microchannel structure, facilitating the mapping of resistivity estimates to physical space.

[0048] For each hexahedral mesh element, a dual-channel state register is established to store the low-frequency resistivity estimate and the high-frequency resistivity estimate, respectively. The low-frequency resistivity estimate can be denoted as... The estimated high-frequency resistivity is denoted as ,in Indicates the first Resistivity estimates of individual hexahedral mesh cells in the low-frequency channel Indicates the first Resistivity estimates of individual hexahedral mesh elements in the high-frequency channel. This represents the sequential numbering of all hexahedral mesh elements. Initially, all... and The nominal resistivity of the gallium-indium alloy at room temperature is set to provide a reasonable starting point, so that the theoretical impedance generated by the forward finite element calculation does not differ too much from the actual measured value, thereby accelerating the convergence of subsequent iterations.

[0049] Select a frequency impedance data frame representing the low frequency from the original multi-frequency impedance spectrum dataset as the low-frequency impedance data frame, and denote it as . Next, select a frequency impedance data frame representing the high frequency as the high-frequency impedance data frame, and denote it as... .in This represents the complex impedance measurement vector for all electrode combinations at the selected low frequency. This represents the complex impedance measurement vector of all electrode combinations at the selected high frequency. The rationale for this is that impedance at low frequencies is more sensitive to temperature and interface effects, while impedance at high frequencies is closer to pure resistive behavior. By reconstructing the low-frequency resistivity field and the high-frequency resistivity field separately, two perspectives with different sensitivities to temperature and strain can be provided for subsequent temperature field extraction and temperature compensation.

[0050] In the frequency domain alternating iterative main loop, the low-frequency channel update phase is executed first. At this point, the current frequency channel is used. The resistivity values ​​of each element in a three-dimensional hexahedral mesh model are used to obtain the theoretical low-frequency impedance vector through finite element forward electric field solution. This can be achieved using operators. The forward solution process for low frequencies is represented by denoted as ,in This represents the theoretical low-frequency complex impedance vector calculated under the current low-frequency resistivity estimation distribution. Indicates that it is composed of all A vector composed of [variables]. Function This represents the forward finite element solution process that maps a three-dimensional resistivity field to boundary impedance measurements. It includes steps such as applying current source boundary conditions to the boundaries of each pair of excitation electrodes, applying current continuity boundary conditions to the unexcited boundaries, and solving the corresponding Poisson equation.

[0051] refer to Figure 4 It contains two parallel two-dimensional resistivity field distribution maps: the left subplot corresponds to the low-frequency resistivity field distribution, and the right subplot corresponds to the high-frequency resistivity field distribution. Each subplot uses a color contour plot, with the horizontal axis representing the spatial position in the X direction (in millimeters) and the vertical axis representing the spatial position in the Y direction (in millimeters). The color bars represent resistivity values ​​in ohm-meters, expressed in scientific notation. The left subplot shows the resistivity field distribution obtained by resistivity tomography reconstruction under selected low-frequency excitation conditions, such as a 1 kHz frequency point. It can be observed from the figure that the resistivity field exhibits a clear non-uniform distribution in space. The region at approximately X = 1 mm and Y = -0.5 mm is marked with a yellow asterisk; the resistivity in this region is significantly higher than the surrounding area, reaching approximately 3.1 x 10^-7 ohm-meters.

[0052] This region of increased resistivity corresponds to the location of temperature increase in the temperature field distribution. Because low-frequency measurements are sensitive to the polarization effect at the interface between the liquid metal and the Ecoflex flexible substrate, as the temperature increases in this region, the degree of interfacial polarization strengthens, and the equivalent resistivity increases accordingly. Furthermore, the region marked with a green square at coordinates of approximately 2.5 mm X and 2 mm Y also exhibits a certain degree of resistivity increase, with a value of approximately 2.95 x 10⁻⁷ ohm-meters. This region corresponds to a strain concentration location, where local tensile strain leads to a decrease in the cross-sectional area and an increase in the length of the microchannel, thus causing the resistivity increase. The low-frequency resistivity field distribution image simultaneously includes the combined effects of temperature and strain on resistivity. The right subplot shows the resistivity field distribution obtained by resistivity tomography reconstruction under selected high-frequency excitation conditions, such as a frequency of 500 kHz. It can be observed from the figure that the overall distribution pattern of the high-frequency resistivity field differs significantly from that of the low-frequency resistivity field. In the temperature rise region where the coordinates are approximately X = 1 mm and Y = -0.5 mm, although the resistivity is still slightly higher than the reference value, the increase is significantly smaller than that in the low-frequency case, and the value is approximately 2.88 x 10^-7 ohm-meters.

[0053] This is because interface polarization does not have enough time to establish under high-frequency measurement conditions, and the influence of temperature on high-frequency impedance is significantly weakened. In contrast, in the strain concentration region at coordinates of approximately 2.5 mm X and 2 mm Y, the high-frequency resistivity shows a more significant increase, reaching approximately 3.05 x 10^-7 ohm-meters, even exceeding the resistivity value of this region under low-frequency conditions. This indicates that high-frequency measurements more prominently reflect the pure resistance changes caused by microchannel geometric deformation, and are more sensitive to strain than to temperature. By comparing the two sub-figures, it is clear that the low-frequency and high-frequency resistivity fields have different sensitivity characteristics to temperature and strain. The temperature-related contribution is more significant in the low-frequency resistivity field, while the strain-related contribution is more prominent in the high-frequency resistivity field. This frequency domain difference is the physical basis for the decoupling of temperature and strain in this invention. In the subsequent temperature compensation processing, by calculating the difference between the low-frequency resistivity estimate and the high-frequency resistivity estimate, the portion of resistivity change mainly caused by temperature can be extracted, thereby achieving the estimation of the temperature field. Meanwhile, by subtracting the temperature-induced resistivity offset from the high-frequency resistivity estimate, we can obtain the pure strain-related resistivity change after eliminating the influence of temperature, providing a reliable data basis for accurately interpreting the triaxial strain components.

[0054] get Afterwards, it can be compared with the actual measurement. By comparison, the low-frequency impedance residual vector is calculated. The residual can be calculated element-wise, denoted as... ,in Indicates the first low frequency Complex residuals of each measurement path Indicates the first low frequency The measured complex impedance of each measurement configuration Indicates the first low frequency The theoretical complex impedance of a measurement configuration. Residual. The larger the value, the greater the deviation between the current three-dimensional resistivity estimate and the true distribution reflected by the measurement.

[0055] To feed the residual information back into the three-dimensional resistivity field, each residual can be processed... Perform a back projection operation. For the first... For each measurement configuration, corresponding to a pair of excitation electrodes and a pair of measurement electrodes, multiple hexahedral mesh elements located near the main current path between the excitation and measurement electrodes can be found in the three-dimensional hexahedral mesh model. These elements are then grouped into a set of main influencing elements. The values ​​are evenly distributed to each hexahedral mesh cell in the main influencing cell set as low-frequency resistivity correction increments, ensuring that resistivity estimation is primarily adjusted in the region that contributes most to the measurement path. This approach projects the error of each measurement path onto the cells corresponding to its physically sensitive region, avoiding meaningless updates to irrelevant areas and improving iteration efficiency.

[0056] After back-projecting the residuals of all measurement paths, each hexahedral mesh cell accumulates the low-frequency resistivity correction increments from multiple measurement paths. The total low-frequency resistivity correction is obtained by summing all correction increments and added to the current value. In the process, the updated low-frequency resistivity estimate is obtained and written back to the first channel of the dual-channel status register. This completes one round of low-frequency channel update.

[0057] The high-frequency channel update phase is similar to the low-frequency channel update phase, except that it uses high-frequency resistivity estimates. and high-frequency impedance data frames The theoretical impedance vector at high frequencies can be obtained through the same finite element forward solution process. The formula is ,in This represents the forward solution operator under high-frequency conditions. Indicates that it is composed of all The resulting vector. Then, the high-frequency residuals are calculated similarly. And update via back projection Since high-frequency measurements are insensitive to interface polarization, their residual updates more accurately reflect changes in the conductive paths related to geometric deformation.

[0058] After updating the low-frequency and high-frequency channels, a total variational smoothing phase is performed. For each hexahedral mesh element, its low-frequency total variational gradient value in the low-frequency channel is calculated. The low-frequency resistivity estimate of the adjacent forward element in the X direction can be denoted as... The low-frequency resistivity estimate of the adjacent backward cell in the X direction is denoted as... The low-frequency resistivity estimates of adjacent forward and backward elements in the Y direction are denoted as follows: and The low-frequency resistivity estimates of adjacent forward and backward elements in the Z direction are denoted as follows: and The low-frequency total variation gradient value of the current hexahedral mesh element can be calculated using the following expression. : .in The vertical bar represents the total variational gradient of the unit in the low-frequency channel. This represents the absolute value operation. and These represent the low-frequency resistivity estimates of the adjacent forward and backward cells in the X direction, respectively. and These represent the corresponding values ​​in the Y direction. and These represent the corresponding values ​​in the Z direction. This expression reflects the total magnitude of the resistivity change of the current cell in the three directions. A large resistivity change indicates a potential significant resistivity jump at that location, corresponding to a boundary or defect in the strain field; if If it is smaller, it may be located in a relatively uniform region.

[0059] Based on all hexahedral mesh elements The sorting process allows selecting the top percentages as low-frequency, high-gradient units, and the rest as low-frequency, flat units. Low-frequency neighborhood mean smoothing is performed only on these flat units to avoid over-smoothing out the true boundaries. In this smoothing operation, the arithmetic mean of the low-frequency resistivity estimates of all neighboring units in the X, Y, and Z directions is taken, and then combined with the currently stored low-frequency resistivity estimates for that flat unit and taken again. The result is then written back to the first channel of that flat unit. This approach preserves both edge information and overall smoothness in the resistivity field, improves iterative stability, and reduces spatial amplification of measurement noise.

[0060] For the high-frequency channel, the high-frequency total variation gradient value can be calculated in the same way. The regular hexahedral mesh element is divided into high-frequency high-gradient elements and high-frequency flat elements. The high-frequency neighborhood mean smoothing operation is only performed on the high-frequency flat elements, and the corresponding smoothing result is written back to the second channel.

[0061] After each round of frequency domain alternation iterations, a convergence criterion can be calculated to determine whether further iterations are needed. The sum of the absolute values ​​of the low-frequency residuals and the high-frequency residuals can be used as the convergence criterion. The expression is ,in Represents the total residual sum of this iteration, with the sign... This represents the summation operation. Indicates the quantity of the measurement configuration. Indicates the first low frequency The residuals of each measurement path, Indicates the first high frequency The residual of each measurement path. When When the resistivity field distribution is less than the preset convergence threshold, it can be considered that the resistivity field distribution is basically consistent with the measured data, and the frequency domain alternating iterative main loop is terminated; when If the convergence threshold is greater than or equal to the threshold, continue to the next iteration. In practice, the maximum number of iterations can be limited to [a certain threshold]. It's my turn Between the wheels, to balance computation time and reconstruction quality.

[0062] refer to Figure 4 , Figure 4This image shows the low-frequency resistivity field distribution obtained by frequency-domain alternating iterative resistivity tomography reconstruction based on total variational constraints. The image uses color contour plots to display the two-dimensional spatial distribution of the resistivity field inside the sensor reconstructed under an excitation frequency of 1 kHz. The horizontal axis represents the position range in the X direction from -4 mm to +4 mm, and the vertical axis represents the position range in the Y direction from -4 mm to +4 mm. The color bars indicate the resistivity numerical range in ohm-meters, expressed in scientific notation as 2.75 x 10⁻⁷ to 3.2 x 10⁻⁷. Red areas represent locations with higher resistivity, and blue areas represent locations with lower resistivity. The contour lines delineate the boundaries of the resistivity gradient. A significant high resistivity region, marked with a yellow pentagram, can be clearly seen near coordinate 1 (comma -0.5). The resistivity in this region reaches approximately 3.1 x 10⁻⁷ ohm-meters, significantly higher than the surrounding background resistivity of approximately 2.8 x 10⁻⁷ ohm-meters. Given that low-frequency measurements are more sensitive to temperature changes, this high resistivity region likely corresponds to a location where the temperature rises inside the sensor. The increased temperature leads to an increase in the resistivity of the liquid metal gallium-indium alloy, while simultaneously enhancing the polarization effect at the interface between the liquid metal and the Ecoflex flexible substrate. The combined effect of these two factors results in a significant increase in the low-frequency resistivity measurement.

[0063] In addition, there is another region of increased resistivity, marked with a green square. This region may correspond to a strain concentration location. Local tensile strain leads to an increase in the length of the microchannel and a decrease in its cross-sectional area, thereby increasing the resistance of the conductive path. The reconstruction process of the low-frequency resistivity field distribution image adopts a frequency domain alternating iterative method. First, low-frequency impedance data frames are extracted from the original multi-frequency impedance spectrum dataset. Then, a three-dimensional hexahedral mesh model of the sensor monitoring area is established. A dual-channel state register is established for each mesh cell to store the low-frequency resistivity estimate and the high-frequency resistivity estimate, respectively. In the iterative main loop, the low-frequency channel update stage performs a finite element forward electric field solution based on the current low-frequency resistivity estimate to obtain the theoretical low-frequency impedance vector. The theoretical value is subtracted from the measured value element by element to obtain the low-frequency impedance residual vector. A back projection operation is performed on each residual to determine the set of main influencing elements, and the residual is evenly distributed to the corresponding element as the resistivity correction increment. Finally, the accumulated correction increment is added to the current resistivity estimate to complete one round of update. In the total variation smoothing stage, the low-frequency total variation gradient value of each grid cell is calculated. Based on the gradient value, the cells are divided into high-gradient cells and flat cells. Only the flat cells are subjected to neighborhood mean smoothing to suppress noise while preserving edge features. After multiple iterations, the loop terminates when the residual converges to a preset threshold, and the final low-frequency resistivity field distribution image is output. The image also shows contour lines represented by thin black lines, which clearly delineate the spatial variation trend of the resistivity gradient. Dense contour lines near temperature-increasing and strain-concentrated regions indicate drastic resistivity changes, while sparse contour lines in the background areas far from these feature regions indicate relatively uniform resistivity. The low-frequency resistivity field distribution image provides important data input for subsequent temperature field extraction and temperature compensation. By comparing it with the high-frequency resistivity field distribution image, temperature and strain effects can be separated, which is a key step in achieving accurate three-dimensional strain measurement.

[0064] In another alternative implementation, the low-frequency impedance data frame and the high-frequency impedance data frame may not be a single frequency point, but rather a weighted average of several low-frequency points and several high-frequency points. That is, multiple low-frequency data frames can be used first. The resistivity field distribution image is reconstructed at several low frequencies, and then weighted and fused from multiple low-frequency reconstructions to form a low-frequency resistivity field distribution image. The same operation is then performed at multiple high frequencies. This further reduces potential random errors or local resonance effects at a single frequency point, making the low-frequency and high-frequency resistivity fields more stable, thus providing a more reliable data foundation for subsequent temperature field extraction and temperature compensation. Regardless of whether a single frequency point or multiple frequency points are fused, both low-frequency and high-frequency resistivity field distribution images are ultimately obtained for subsequent steps to calculate temperature compensation and triaxial strain components.

[0065] In calculating temperature field distribution data, it is first necessary to clarify the correspondence between the low-frequency resistivity field distribution image and the high-frequency resistivity field distribution image in terms of data structure. During reconstruction, the monitoring area has been divided into multiple hexahedral grid cells, each with a unique three-dimensional grid index, which can be denoted as... ,in, This indicates the layer number of the hexahedral mesh element in the X direction. This indicates the layer number in the Y direction. This indicates the layer number in the Z direction. Typically, each hexahedral mesh element is then sequentially numbered. This makes it easier to store all hexahedral mesh elements in a one-dimensional array within the program. For the element numbered... The resistivity estimate of a regular hexahedral mesh element in a low-frequency resistivity field distribution image can be denoted as: The resistivity estimate in the high-frequency resistivity field distribution image can be denoted as: ,in, This represents the estimated low-frequency resistivity obtained by reconstruction under selected low-frequency conditions. This represents the estimated high-frequency resistivity obtained by reconstruction under selected high-frequency conditions.

[0066] Within the same hexahedral mesh cell, the geometry of a liquid metal microchannel network is identical, but the equivalent resistivity measured at different frequencies exhibits varying degrees of temperature sensitivity. Low-frequency measurements incorporate more effects from interfacial polarization, ion migration, and slow polarization within the material, which change more significantly with temperature. High-frequency measurements, on the other hand, are less sensitive to these slow processes, more closely resembling the characteristics of a simple conductive path. Therefore, under strain-free conditions, changing only the temperature typically results in a greater change in low-frequency resistivity than in high-frequency resistivity. Based on this behavior, temperature can be estimated by the difference between the low-frequency and high-frequency resistivity field distribution images.

[0067] Specifically, for each hexahedral mesh element, its frequency domain resistivity difference can be calculated first. The frequency domain resistivity difference can be denoted as... The calculation expression is: ,in This represents the difference between the low-frequency resistivity estimate and the high-frequency resistivity estimate of the current hexahedral mesh element. Since low frequencies are more susceptible to temperature-induced polarization, this difference often occurs. Greater than Therefore It is usually a positive value, and the larger the value, the more obvious the temperature at that location deviates from the baseline state.

[0068] Before practical use, calibration tests are needed to determine the temperature response of the liquid metal microchannel network. Specifically, this can be achieved by placing a three-dimensional flexible distributed strain sensor in a constant-temperature chamber under strain-free conditions and performing measurements at multiple different constant temperatures. For example, selecting... , , , At isothermal points, low-frequency and high-frequency resistivity field distribution images were acquired for each temperature point, and the corresponding values ​​were calculated for each hexahedral mesh element. Based on this, the relationship between the frequency domain resistivity difference and temperature deviation can be fitted. For ease of online calculation, this relationship can be approximated as a linear relationship, in the form of: ,in This represents the coefficient of resistivity change per unit temperature. This represents the estimated temperature deviation of the current hexahedral mesh element. It is the coefficient of resistivity change per unit temperature. This can be understood as each change in temperature When, under strain-free conditions, the average change in frequency domain resistivity difference is usually expressed as As a unit. In many scenarios, it can be... It can be considered a global constant, or different values ​​can be set for different regions based on the calibration results. .

[0069] When running online, obtain the current After that, based on the calibration results Estimate the temperature deviation. The estimated temperature deviation can be denoted as: The calculation expression is: ,in This indicates the temperature deviation of the current hexahedral mesh element relative to the ambient reference temperature value, molecule This represents the frequency domain resistivity difference of the unit, and the denominator is... This represents the coefficient of resistivity change per unit temperature. The advantage of this approach is that by utilizing the difference between low and high frequencies, the components related to absolute geometric deformation can be largely canceled out, highlighting the changes caused by temperature, which is equivalent to constructing a "temperature channel".

[0070] By analyzing the temperature deviation from the estimated value, the absolute temperature can be obtained. The environmental reference temperature value can be denoted as... ,in This indicates the reference temperature of the sensor's environment during calibration or initialization, for example... The absolute temperature estimate can be denoted as... The calculation expression is: ,in This represents the estimated absolute temperature of the current hexahedral mesh element. When A positive value indicates that the temperature at that location is higher than the ambient reference temperature. A negative value indicates that the temperature at that location is lower than the ambient reference temperature. This applies to all hexahedral mesh elements. Arranged according to their respective three-dimensional grid indices, the temperature field distribution data is formed. The temperature field distribution data is stored in the form of a three-dimensional array, which can be directly displayed on the visualization interface as a pseudo-color image, or used as input for subsequent temperature compensation calculations.

[0071] To separate the temperature effect from the high-frequency resistivity field distribution image, a temperature-induced resistivity offset needs to be constructed. This temperature-induced resistivity offset can be denoted as... And adopt a linear relationship ,in This represents the resistivity shift of the current hexahedral mesh element due to temperature deviation. This indicates that the temperature of the unit deviates from the estimated value. It still represents the coefficient of resistivity change per unit temperature. Because... and The definition originates from the calibration process, and is constructed in this way. There is a one-to-one correspondence between this and the actual temperature-induced change in resistivity. When executed online, although mathematically... and An approximate correspondence exists, but it cannot be calculated explicitly. and This makes the implementation process clear and controllable, for example, it can be done in Upper and lower limit constraints are added to avoid unreasonable temperature estimates.

[0072] After obtaining the temperature-induced resistivity offset, it can be subtracted from the high-frequency resistivity estimate to obtain the temperature-compensated resistivity value. This temperature-compensated resistivity value can be denoted as... The calculation expression is: ,in This represents the resistivity value remaining after the temperature effect has been subtracted from the current hexahedral mesh element. This represents the estimated high-frequency resistivity of the element in the high-frequency resistivity field distribution image. This represents the temperature-induced resistivity offset. The reason for choosing a high-frequency resistivity estimate as the benchmark is that high-frequency measurements are inherently less sensitive to temperature-dependent polarization; after temperature correction, the resulting value... It more closely approximates the change in conductive path caused purely by geometric deformation and strain. This property directly serves subsequent strain interpretation.

[0073] In one alternative implementation, after obtaining the temperature field distribution data, the data can be further processed... Perform spatial smoothing or outlier removal. For example, for each hexahedral mesh element, take the values ​​of several neighboring elements in the X, Y, and Z directions. The median, replacing the original This filters out local noise, making the temperature field distribution data smoother and preventing single-point temperature estimation errors from having an excessive impact on the temperature-compensated resistivity field distribution data. Similarly, it can also be used for... Perform a light smoothing to reduce high-frequency noise introduced by reconstruction errors.

[0074] After calculating the temperature field distribution data and the temperature-compensated resistivity field distribution data, the three-dimensional strain information can be deduced based on the temperature-compensated resistivity field distribution data. Specifically, for each hexahedral mesh element, the temperature-compensated resistivity value needs to be compared with the initial nominal resistivity value to first obtain the net resistivity change, then convert it into the relative resistivity change rate, and finally obtain the X-axis strain component, Y-axis strain component, and Z-axis strain component respectively based on the pre-calibrated X-axis strain sensitivity coefficient, Y-axis strain sensitivity coefficient, and Z-axis strain sensitivity coefficient.

[0075] The nominal resistivity value can be denoted as: ,in The reference resistivity of the liquid metal microchannel network under reference temperature and zero strain conditions can be obtained by averaging multiple measurements taken by a three-dimensional flexible distributed strain sensor under constant temperature and no-load conditions. For a specific hexahedral mesh element, its temperature-compensated resistivity is... First, calculate the change in net resistivity. The change in net resistivity can be denoted as... The calculation expression is: ,in This represents the change in resistivity of the hexahedral mesh cell relative to its nominal resistivity after the temperature effect has been removed. If... A positive value indicates that the equivalent resistivity at that location is higher than the reference state, which usually corresponds to local stretching or a reduction in the conductive cross-section; a negative value indicates that the equivalent resistivity is lower than the reference state, which may correspond to local compression or an enhanced conductive path.

[0076] In many conductive composite materials, a relatively stable, approximately linear relationship exists between the relative change in resistivity and strain. To convert the net resistivity change into a dimensionless proportionality, the relative resistivity change rate can be calculated. The relative resistivity change rate can be denoted as... The calculation expression is: ,in The numerator represents the rate of change of relative resistivity of the current hexahedral mesh element. The denominator represents the change in net resistivity. This indicates the nominal resistivity value. It can be regarded as "the proportion of change in electrical conductivity at this location relative to the initial state when there is no temperature effect", which is the unified input for subsequent strain interpretation.

[0077] Next, the relative resistivity change rate needs to be mapped to strain components in three orthogonal directions. To do this, uniaxial strain can be applied during sensor manufacturing and calibration, and the corresponding relative resistivity change rate can be measured to obtain the X-axis strain sensitivity coefficient, Y-axis strain sensitivity coefficient, and Z-axis strain sensitivity coefficient. The X-axis strain sensitivity coefficient can be denoted as... The strain sensitivity coefficient in the Y direction is denoted as The Z-direction strain sensitivity coefficient is denoted as ,in This represents the proportional relationship between the rate of change of relative resistivity and the strain in the X direction when only strain is applied in the X direction. This indicates the proportional relationship when only the Y-direction strain is applied. This indicates the proportional relationship when only Z-direction strain is applied. , and It can be obtained through linear fitting, so that it can be solved by simple division when used online.

[0078] For a given hexahedral mesh element, the rate of change of relative resistivity is obtained. Then, the X-direction strain components, Y-direction strain components, and Z-direction strain components can be calculated using the following formula. The X-direction strain component is denoted as... The strain component in the Y direction is denoted as The Z-direction strain component is denoted as The calculation expression is: , , ,in, This represents the X-axis strain component of the current hexahedral mesh element. This represents the Y-axis strain component of the current hexahedral mesh element. This represents the Z-axis strain component of the current hexahedral mesh element. Denominator , , These are the strain sensitivity coefficients for the corresponding directions. This approach is equivalent to assuming that, after temperature compensation, the relative change in resistivity can be explained by the different sensitivities of strain in the three directions to the same scalar. Different strain sensitivity coefficients can reflect the structural differences and conductive path distribution differences of the sensor in the three directions. For example, a microchannel array in the X-direction may be more susceptible to a certain type of load than a microchannel array in the Y-direction. and It will be different.

[0079] refer to Figure 2 The graph contains three parallel two-dimensional field distribution maps, showing the spatial distribution characteristics of the extracted temperature field, the temperature-affected mixed state, and the temperature-compensated pure strain state from left to right. The leftmost sub-graph shows the temperature field distribution data extracted from the frequency domain difference between the low-frequency and high-frequency resistivity field distribution images. This graph uses a heatmap format, with the horizontal axis representing the spatial location in the X direction (in millimeters) and the vertical axis representing the spatial location in the Y direction (in millimeters). The color bars use a warm color scheme, with dark red areas indicating higher temperatures and yellow areas indicating moderate temperatures. A significant temperature peak can be observed in the region at approximately X = 3 mm and Y = 7 mm, where the temperature is about 10 degrees Celsius higher than the ambient reference temperature.

[0080] Meanwhile, another temperature rise zone exists in the area at approximately X = 7 mm, Y = 3 mm, with the temperature about 5 degrees Celsius higher than the baseline value. This non-uniformity in temperature distribution may originate from local heating by an external heat source or local temperature rise caused by strain work done by the sensor during operation. The middle subplot shows the relative resistivity change rate distribution calculated directly from the high-frequency resistivity field distribution image before temperature compensation. This plot also uses a two-dimensional color field plot, with the horizontal axis representing the X direction and the vertical axis representing the Y direction. The color bars represent the percentage change in resistivity relative to the nominal value. As can be seen from the figure, there is a significant peak area of ​​relative resistivity change in the central region of the sensor, near the coordinates of approximately X = 5 mm, Y = 5 mm, with a relative change rate of about 2.3%. This change includes both the effect of microchannel geometry changes caused by actual strain on resistivity and the contribution of the temperature coefficient effect of liquid metal resistivity caused by temperature changes. Since the temperature field distribution shows that there is indeed a certain temperature deviation in this area, the relative resistivity change shown in the middle subplot is actually the result of the superposition of temperature and strain effects.

[0081] The rightmost subplot shows the distribution of the relative rate of change of resistivity after temperature compensation. In this figure, by subtracting the temperature-induced resistivity offset calculated from the temperature field from the high-frequency resistivity estimate, the influence of temperature change on resistivity is effectively eliminated. The remaining relative rate of change mainly reflects the change in the conductive path caused purely by strain. It can be observed from the figure that the distribution of the relative rate of change of resistivity after compensation is more concentrated in the central region of the sensor, with the peak position basically consistent with that before compensation, but the value is slightly lower, approximately 1.5%. Comparing the two subplots before and after compensation, it is clear that the temperature compensation operation successfully separates the temperature-induced resistivity change from the total change, making the field distribution after compensation more accurately reflect the true strain state. These three subplots clearly demonstrate the complete process of temperature field extraction, mixed state identification, and temperature compensation processing from left to right, verifying the effectiveness of the temperature estimation method and temperature compensation strategy based on frequency domain resistivity difference proposed in this invention.

[0082] All hexahedral mesh elements Arranging the data according to the three-dimensional mesh index yields the strain field distribution in the X-axis; [then, all of these are processed]. Arranging the data according to the three-dimensional mesh index yields the strain field distribution in the Y direction; [then, all of these are processed]. Arranged according to the 3D mesh index, the Z-axis strain field distribution is obtained. The three strain field distributions can be stored as 3D scalar fields and displayed through cross-sectional plots, isosurfaces, or volume rendering. For example, a fixed Z-axis layer can be selected to display the X-axis strain field distribution of the corresponding layer, which can be used to observe the tension and compression of the subgrade along the track direction at that depth; multiple cross-sections can also be taken along the X direction to view the Y-axis strain field distribution and evaluate the lateral deformation of the subgrade.

[0083] In an alternative implementation, if it is desired to correct for nonlinearities in strain interpretation, a lookup table can be constructed during the calibration phase instead of simply using a single one. , and For example, different linear intervals can be fitted for different strain ranges to form a piecewise linear mapping. During online operation, first, based on the current... Select the corresponding coefficient group within the given range, and then calculate. , and This allows for improved interpretation accuracy over large strain ranges while maintaining moderate implementation complexity.

[0084] refer to Figure 3 , Figure 3This diagram contains three parallel two-dimensional strain field distribution plots, corresponding from left to right to the X-axis, Y-axis, and Z-axis strain field distributions, respectively. Each sub-plot uses a color contour plot, with the horizontal axis representing the spatial location in the X-axis (in millimeters), the vertical axis representing the spatial location in the Y-axis (in millimeters), and the color bars representing strain values ​​in microstrain. The leftmost sub-plot shows the X-axis strain field distribution. This plot uses a red-blue two-tone color scheme, where red areas represent tensile strain and blue areas represent compressive strain.

[0085] As can be observed from the figure, a significant tensile strain concentration zone exists in the central region of the sensor, approximately at coordinates X = 5 mm, Y = 5 mm, with the X-direction strain reaching approximately 12,000 microstrains. This indicates that the sensor at this location experiences a relatively significant tensile load along the X direction, leading to an increase in the channel length and a decrease in the cross-sectional area of ​​the X-direction microchannel array, thereby increasing the equivalent resistance of the conductive path in the X direction. Furthermore, periodic strain fluctuations along both the X and Y directions can be observed in the figure. These fluctuations may originate from differences in strain distribution at different locations of the sensor under bending or torsion conditions. The middle subfigure shows the Y-direction strain field distribution. It can be seen from the figure that the distribution characteristics of the Y-direction strain field differ significantly from those of the X-direction strain field. A compressive strain zone appears in the region at coordinates approximately X = 3 mm, Y = 7 mm, with a Y-direction strain of approximately -5,000 microstrains, indicating that this location is subjected to a compressive load along the Y direction. Simultaneously, a tensile strain zone appears in the region at coordinates approximately X = 7 mm, Y = 3 mm, with a Y-direction strain of approximately +8,000 microstrains.

[0086] The spatial separation of tensile and compressive strains indicates that the sensor may be under complex multiaxial stress, with different regions of the Y-axis microchannel array subjected to loads in different directions. The rightmost subfigure shows the Z-axis strain field distribution. Z-axis strain typically corresponds to the sensor's deformation in the thickness or vertical direction. The figure shows a Z-axis tensile strain zone at the sensor's center, with a value of approximately 6000 microstrain, while Z-axis compressive strain appears in some areas near the edges, with a value of approximately -3000 microstrain. This spatial distribution characteristic of Z-axis strain may be related to the thickness deformation caused by the Poisson effect when the sensor is subjected to in-plane tension or compression, or it may reflect the normal load acting on the sensor in practical applications. By comparing the strain field distributions in the three directions, significant differences in the spatial distribution and magnitude of the X-axis, Y-axis, and Z-axis strains can be observed. This fully demonstrates that the three-dimensional flexible distributed strain sensor of this invention can independently sense strain components in three orthogonal directions, achieving true three-dimensional strain measurement. Figure 3The triaxial strain field distribution results shown were obtained after temperature compensation. Therefore, these strain values ​​truly reflect the mechanical load state of the sensor, without including the interference of temperature changes.

[0087] The present invention has been described in detail above. Specific examples have been used to illustrate the principles and implementation methods of the invention. The descriptions of the embodiments above are merely for the purpose of helping to understand the method and core ideas of the present invention. It should be noted that those skilled in the art can make various improvements and modifications to the present invention without departing from its principles, and these improvements and modifications also fall within the protection scope of the claims of the present invention.

Claims

1. A three-dimensional flexible distributed strain sensor, characterized in that, The system includes a sensor body and a signal conditioning module. The sensor body comprises an Ecoflex flexible substrate, a liquid metal microchannel network, and a ring electrode array. The Ecoflex flexible substrate contains a three-dimensional interconnected microchannel structure formed using a 3D printing sacrificial template process. The liquid metal microchannel network is composed of a gallium-indium alloy filled within the three-dimensional interconnected microchannel structure. The liquid metal microchannel network includes X-direction microchannel groups extending along the X direction, Y-direction microchannel groups extending along the Y direction, and Z-direction microchannel groups extending along the Z direction. The X, Y, and Z directions are orthogonal to each other. The X-direction microchannel groups and Y-direction microchannel groups... The Z-axis microchannel group is interconnected at its spatial intersection; the ring electrode array is made of a flexible printed circuit board and includes multiple electrode units uniformly distributed along the circumference of the Ecoflex flexible substrate, with each electrode unit electrically connected to a corresponding port of the liquid metal microchannel network; the signal conditioning module includes a direct digital frequency synthesis signal source, a current injection switching matrix, and a lock-in amplifier detector; the direct digital frequency synthesis signal source is used to generate a sweep frequency excitation signal; the current injection switching matrix is ​​connected to all electrode units of the ring electrode array; the lock-in amplifier detector is used to detect the voltage response signal and extract amplitude and phase information.

2. The three-dimensional flexible distributed strain sensor according to claim 1, characterized in that, The ring electrode array comprises thirty-two electrode units.

3. The three-dimensional flexible distributed strain sensor according to claim 1, characterized in that, The sweep excitation signal generated by the direct digital frequency synthesis signal source covers the low-frequency to high-frequency bands.

4. The three-dimensional flexible distributed strain sensor according to claim 1, characterized in that, The current injection switching matrix is ​​used to switch the combination of excitation electrode pairs and measurement electrode pairs according to the pattern of injecting current into adjacent electrodes and measuring voltage on the opposite electrode.

5. A strain interpretation method based on temperature compensation using a three-dimensional flexible distributed strain sensor according to any one of claims 1 to 4, characterized in that, Includes the following steps: Step 1: Generate a sweep excitation signal using a direct digital frequency synthesis signal source, and complete the measurement by sequentially traversing all electrode combinations using a current injection switching matrix. Collect the voltage response amplitude and voltage response phase using a lock-in amplifier detector and calculate the complex impedance measurement value to obtain the original dataset of the multi-frequency impedance spectrum. Step 2: Perform frequency-domain alternating iterative resistance tomography reconstruction processing based on total variation constraints on the original multi-frequency impedance spectrum dataset to output low-frequency resistivity field distribution images and high-frequency resistivity field distribution images; Step 3: Calculate the temperature field distribution data and the temperature-compensated resistivity field distribution data based on the low-frequency resistivity field distribution image and the high-frequency resistivity field distribution image; Step 4: Calculate and output the X-axis strain field distribution, Y-axis strain field distribution, and Z-axis strain field distribution based on the temperature-compensated resistivity field distribution data.

6. The strain interpretation method for temperature compensation according to claim 5, characterized in that, In step one, the frequency sweep excitation signal gradually increases from the starting frequency to the ending frequency. After measuring all electrode combinations at each excitation frequency, all complex impedance measurements corresponding to the current excitation frequency are stored as a frame of frequency impedance data. The frequency impedance data of all excitation frequencies constitute the original dataset of the multi-frequency impedance spectrum.

7. The strain interpretation method for temperature compensation according to claim 5, characterized in that, In step two, the frequency domain alternating iterative resistance tomography reconstruction process includes: establishing a three-dimensional hexahedral mesh model for the monitoring area of ​​the sensor body, dividing it into multiple hexahedral mesh units at equal intervals along the X, Y, and Z directions, and assigning a three-dimensional mesh index to each hexahedral mesh unit; establishing a dual-channel state register for each hexahedral mesh unit, which includes a first channel for storing low-frequency resistivity estimates and a second channel for storing high-frequency resistivity estimates; extracting low-frequency impedance data frames and high-frequency impedance data frames from the original multi-frequency impedance spectrum dataset; and entering the frequency domain alternating iterative main loop, each round of which includes a low-frequency channel update stage, a high-frequency channel update stage, and a total variational smoothing stage executed sequentially.

8. The strain interpretation method with temperature compensation according to claim 7, characterized in that, The execution process of the low-frequency channel update stage is as follows: Based on the estimated low-frequency resistivity values ​​of all hexahedral mesh elements, the finite element forward electric field solution is performed to obtain the theoretical low-frequency impedance vector; the theoretical low-frequency impedance vector is subtracted element by element from the corresponding measured low-frequency impedance vector in the low-frequency impedance data frame to obtain the low-frequency impedance residual vector; a back projection operation is performed on each residual element in the low-frequency impedance residual vector to determine the set of main influencing elements corresponding to each residual element; the value of each residual element is evenly distributed to each hexahedral mesh element in the corresponding set of main influencing elements as the low-frequency resistivity correction increment. For each hexahedral mesh cell, the total low-frequency resistivity correction is obtained by summing all the received low-frequency resistivity correction increments. The total low-frequency resistivity correction is then added to the low-frequency resistivity estimate of the current hexahedral mesh cell and written back to the first channel. In the high-frequency channel update stage, the high-frequency resistivity estimate in the second channel is updated in the same way as in the low-frequency channel update stage.

9. The strain interpretation method with temperature compensation according to claim 7, characterized in that, The execution process of the total variation smoothing stage is as follows: calculate the low-frequency total variation gradient value for each hexahedral mesh cell. The low-frequency total variation gradient value is the sum of the absolute values ​​of the low-frequency resistivity difference values ​​of the current hexahedral mesh cell in the X, Y and Z directions. Based on the low-frequency total variation gradient value, all regular hexahedral mesh elements are divided into low-frequency high-gradient elements and low-frequency flat elements. For each low-frequency flat cell, a low-frequency neighborhood mean smoothing operation is performed. The arithmetic mean of the low-frequency resistivity estimates of all neighboring cells of the current low-frequency flat cell is taken as the arithmetic mean of the low-frequency resistivity estimates currently stored in the current low-frequency flat cell and then written back to the first channel. The same smoothing operation is performed on the high-frequency resistivity estimates in the second channel.

10. The strain interpretation method for temperature compensation according to claim 5, characterized in that, In step three, the following operations are performed on each hexahedral mesh cell: the low-frequency resistivity estimate of the current hexahedral mesh cell is subtracted from the high-frequency resistivity estimate of the current hexahedral mesh cell to obtain the frequency domain resistivity difference; the frequency domain resistivity difference is divided by the unit temperature resistivity change coefficient to obtain the temperature deviation estimate; the temperature deviation estimate is added to the ambient reference temperature to obtain the absolute temperature estimate; the high-frequency resistivity estimate of the current hexahedral mesh cell is subtracted from the product of the temperature deviation estimate and the unit temperature resistivity change coefficient to obtain the temperature-compensated resistivity value.

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